Although this would apparently earn them animosity with the owners of the Jennie June copyright, this business tactic seems to have worked pretty well. Lily/Miss Midget shows up in the most states. Her most common appearances seem to be in Kansas, Iowa, Illinois, Missouri, and Oklahoma. Jennie June shows up mostly in Ohio and Pennsylvania, and otherwise in the most northern states. I found about ten with no name given in promotional material.

However, although Lillie shows up into the 1950s, I have never seen any inkling of a surviving script. (Maybe they should have been freer about sharing it.) Everything I know is from old articles. Jennie June was published first, she is the one with a surviving script that is available today, and there are newspaper articles on Jennie June weddings up to the 1970's and 80's. 
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People clearly worked their own thing sometimes. The cast might be full of celebrities, or cartoon and nursery rhyme characters.
There seems to have been some confusion; a fictional bride might show up alongside Minnie Warren and Commodore Nutt, real people from the real wedding. On at least one occasion, in 1957, the bride was called Lavinia Warren.

In a Jennie June production in the 1920's, Minnie Warren was be the maid of honor and Commodore Nutt the groomsman; Reverend Tie-em-up officiated; Bo Peep and Miss Muffet were bridesmaids, and guests had names like Mr. and Mrs. Barney Google or the Katzenjammer Kids (from comic strips) or Mr. and Mrs. Simon Says. Another had some of the same guests, but featured the Fairy Queen and President Coolidge in attendance, and a “rejected suitor” named Percival Doolittle (rather than Commodore Nutt). One promotion in Chillicothe, Missouri in 1926 mentioned Jennie June Midget’s sister, Mrs. Sam Little.

Lillie Putian weddings do not have cast lists quite as elaborate, as far as I could see, but seemed to feature an awful lot of cousins who sang solos. A 1914 Lillie wedding had a much more simplistic cast list, with the newspaper article merely mentioning the maid of honor, best man, flower girls, various family members, three old maids, and in its one confused nod to real life, Colonel Nutt as a rejected suitor.
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In both Lily productions and Jennie productions, there were some common popular songs that showed up: “Oh, Promise Me,” “I Love You Truly,” “You are the Ideal of My Dreams,” “I'd Love to Live in Loveland With a Girl Like You,” “When You and I Were Young, Maggie,” “I Cannot Sing the Old Songs” and “Silver Threads Among the Gold” (these two usually by Grandma and Grandpa Thumb), “When You and I were Young,” and “When You Look in the Heart of a Rose.” 

I'm still hoping maybe someone somewhere has a copy of the May Burnworth script, or more history on her. That story, about the woman traveling around America directing pageants, sounds amazing.
Who was C. A. Rose, anyway? Was it her? ... Well, probably not.

When I researched Tom Thumb weddings a while back, there was one thing that mystified me - and that was the name Jennie June. This was the name used in  the Baker's play published in 1898, and in the majority of advertisements and reviews found as my research.

It was clear how the Tom Thumb name had been inherited from General Tom Thumb and thence from the fairytale, but who was Jennie June? Wouldn't it have made more sense to call the bride Lavinia, after Mrs. Stratton? Another skit gave the bride the thematically appropriate name Lilly Putian.

Maybe I'm overthinking it and it's just a random name that they threw in thinking it sounded good. Maybe, as I initially guessed, there's a connection to the tradition of June weddings. There have been plenty of women with this name. ​Anyway, I set out to create a timeline, seeking out possible inspirations for the name of Tom's bride.

1853: A book of poems by Benjamin Franklin Taylor, "January and June." I'm not sure of the original date of publication of these poems; they may have been previously published in the Evening Journal.
Anyway, one poem is titled Jenny June/The Beautiful River.
In a twilight like that, Jenny June for a bride,
​Oh ! what more of the world could one wish for beside,

Jennie June (Jane Cunningham Croly): A very famous journalist who founded the Sorosis club for women in 1868. Wikipedia mentions that she may have first used the pen name of Jennie June as early as 1855. Whenever she first used it, she seems to have used a few at first before settling on that one and starting a trend of alliterative pen names.
Her pen name soon became a household name, with her columns and clubs gaining popularity. In the 1880's, she edited a series of manuals for ladies, including an American Cookery Book and several books on needlework and sewing.
I found many different accounts of what inspired her name.

• ​The New England Magazine, Volume 17 quotes Croly as saying that the nickname came from her childhood and was given to her by Reverend W. W. King, pastor of her Poughkeepsie parish. "He found three verses inscribed to ‘Jennie June’ in a volume of poems, which he sent to me with this poem marked and dedicated to me. The name was not used in the family… it was only used occasionally by an elder brother."
• In 1900, the Oshkosh Daily Northwestern from Oshkosh, Wisconsin recounts that it was a book of poems by Benjamin F. Taylor, a gift from her Unitarian pastor, with a remark of 'These are for the Juniest little girl that I know.'" The article mentions both "Jenny June" and "January," strongly indicating that this volume was January and June. In this account, Jane was known for a long time to her friends by that nickname, and remembered it when she was choosing a pen name.
• In the December 1899 Detroit Free Press from Detroit, Michigan and in the April 1900 Intermountain Catholic from Salt Lake City, Utah: When she was twelve, "a gentleman who had been visiting her family wrote to a friend: "She Is the Juniest little girl I ever knew!"
• This blog post says that it was a poem by Benjamin F. Taylor, sent to her by her Poughkeepsie pastor when she was twelve, with the name underlined and a comment of "You are the Juniest little girl I know." 
• An Inkwell of Pen Names by Stephen Smith (2006) claims that "In 1855 she found the name Jennie June in a poem by Benjamin Franklin Taylor."

In all of these, there are slightly different details. It's always an older gentleman friend, usually her pastor, who gives her a book of poems (by Taylor, we're told) and/or the sobriquet of "the Juniest little girl I know."
However, Croly would have been twelve in 1841, and the earliest I can find proof of Taylor's poem is 1853. At seventy, when she is supposed to have been interviewed about this, Croly may have had trouble remembering the real specifics. However, the varying accounts, as well as the overly flowery style in the Oshkosh paper, make me think that there was some significant embellishment going on. Also, the first-person account in the New England Magazine never mentions Taylor.

1863:  – “Jennie June” appears in Beadle's Dime Song Book, copied by permission of Firth, Son & Co. “Did you see dear Jennie June . . .”

Also in 1863, General Tom Thumb got married, and the wedding was a huge media spectacle. In following years, other small performers, like Francis Joseph Flynn/General Mite in 1884, would also have widely publicized weddings. In 1892, of performer Admiral Dot and his wife Lottie Swartwood, it was said that "in their wedding garments they looked more like pretty little children than like a man and woman about to embark on the uncertain sea of matrimony." Perhaps it wasn't just the Strattons who inspired the wedding pageant.​

1875-1876: The McLoughlin Brothers Paper Dolls includes a doll named Jennie June - a small girl with several different outfits, sold alongside characters like Polly Prim and Gerty Good for 8 cents; these were part of a series of smaller paper dolls, 5 5/8 " inches tall. 
I don't know exactly when this paper doll first appeared; the Uniform Trade List Circular has a mention of the name dated in 1866.
Interestingly, there were also McLoughlin paper doll versions of General Tom Thumb, Lavinia Stratton, and their companions.

1891: The absolute earliest mention I've found of a Tom Thumb wedding in print - hosted by the First African Presbyterian Church. William Dorsey’s Philadelphia cites the Leon Gardiner Collection of the Historical Society of Pennsylvania.

1892: A song called “Little Jennie June” is printed in Album Melodies by Richard Ferber.

1896: “Sweet Jennie June,” a song by Henry J. Sayers

1898: Thirty-five years after the Stratton wedding, The Walter H. Baker & Co. of Boston put out a play 35 years later called, 'The Tom Thumb Wedding" - “as originally performed at the Union Tabernacle Church, Philadelphia, PA”. From this, we know that there was already a pageant tradition forming; it was just that now people were creating scripts. 
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Here, the bride is named Jennie June, and this was the name I found most often browsing through old newspapers and photos. This was the name that kicked off a burning question that would only ever torment one single person in history.

However, at the time, there were other scripts bopping around. Jennie June's chief rival seems to have been Lillian Putian Midget or simply Miss Midget. Again, this name seems less random than Jennie June. People would have gotten the reference to Gulliver's Travels; Tom Thumb was often compared to the people of Lilliput, and at their wedding, General Tom Thumb and his bride were headlined as  "The Loving Lilliputians." (Speaking of General Tom Thumb, in Ohio in 1957, there was a "Mock Marriage of Tom Thumb and Miss Lavinia Warren."

​The earliest mention of Lillian I could find dates to 1901 in the Oklahoman, with a facetious wedding announcement. The writer was clearly having fun with references, as there's a mention of their address at "Gullivar Avenue."

In 1911, a copyright was issued for "The Marriage of Miss Midget," or "The Marriage of the Midgets, or The Tom Thumb Wedding," written by May Burnworth. It was renewed in 1914. This was the Lillian Putian version - not the Bakers Plays version, though it usually bore the same name. Here, in a brochure filled with glowing reviews, at the end there is a stern "WARNING!!" All public performances must be under the direction of C. A. Rose, of the Baxter Printing Company, Kansas City, MO.

(Incidentally, I found a mention in an 1914 Illinois Newspaper -

• "The Dorcas society expect to present The Marriage of Miss Midget or Tom Thumb's Wedding, at the opera house on Tuesday evening , Feb. 3 . This entertainment is given by about forty children between the ages of three and twelve years who will be drilled by a lady director sent by C. A. Rose of Kansas City who has had charge of this entertainment for a number of years and who is making a big success of it.")
• And another here, with C. A. Rose once again sending a director to oversee the play.

​Still, that warning is pretty strong. In this reprint of their play, Bakers' puts up a bit of a defense.

• "Persons in Jacksonville, Florida, and in Kansas City, Missouri, who put out similar entertainments under the titles, "The Marriage of the Tots," "The Jennie June Wedding" and " The Marriage of the Midgets, or the Tom Thumb Wedding” have been calling the attention of our customers to what thy describe as an “infringement” of their “rights” . . . citing “copyrights issued in 1911 and 1914, thirteen to sixteen years later than the date under which we claim. Such a claim is, of course, mere nonsense.” 

Shots fired! There's a clear reference to C. A. Rose with the "Kansas City, Missouri" and the copyright dates. However, the other references (such as Florida) are still mystifying to me. It appears there were other scripts floating around that have since been lost. It also appears that Bakers' Plays weren't the only people using the Jennie June name. Wherever she came from, she wasn't copyrighted. Could the name have been traditional to the pageant even before the script was published?

In the meantime, there were some odd blends of the two brides' names. In an announcement in 1915, Mr. and Mrs. Lyttle Smalle Lilliput announced their daughter Jennie June's wedding. And in 1926, the bride was Miss Jennie June Midget.
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However, ultimately it seems to me that the Baker play and Jennie June have lasted longer. At least, in this day and age, I can easily track down the Baker play online, while I'm at a loss for finding any others. And references abound - for instance, there was a skit called "Tom Thumb's First Wedding Anniversary" by Donald V. Hock, published in 1934, with Jennie as the wife.

Anyway, back to the mystery at hand. Namely - who is Jennie June?

Maybe the writers thought of the little paper doll; maybe it was inspired by someone's copy of a Jennie June manual for ladies; perhaps they remembered Taylor's poem with the line "Jennie June for a bride." But I'm guessing it was just a random name that the writer found cute - most likely something born in the original Union Tabernacle Church skit cited by Baker's Plays.