"Hugo's Waiting" by Frances Essex, Davenport Gazette 1871

The first published story by Alice French was published in the Davenport Gazette in February 1871 under the pseudonym Frances Essex. In the only currently existing biography of French, George McMichael's 1965 Journey to Obscurity: The Life of Octave Thanet, McMichael notes "The manuscript remained among her papers after her death. Her mother had written on it 'This is Alice's first published story.'" (note on page 43). 

I also send that little story I read so long ago to you. . .Do you remember it? Mr. Sanders wanted something for this Saturday's Gazette, so I finished up that. It is very stupid, I know, but it was written for the popular taste, which is also stupid.
--Alice French to Anna Dawes, February 4, 1871

While the manuscript is among French's papers, her statements about that story clearly indicate why I can't find a digital copy today. McMichael suggests that the paper likely published it because the editor "was a friend of Alice's father and knew the virtues of local authors" (McMichael, 43).

You can check out the biography, which is a published version of McMichael's dissertation, at the Internet Archive if you have an account.