In the one instance in which this word occurs in Jane Austen’s letters it doesn’t mean what the spelling appears to suggest (innermost?):

he said the fleas were so starved when he came back from Chawton that they all flew upon him and eenermost eat him up (letter 93).

The quotation is from a part of the letter which was written by Jane Austen’s 13-year-old niece Elizabeth, and the word seems to mean “almost”. Unfortunately, I couldn’t verify this in the OED, as it doesn’t include the word as an entry.

Having a look at what we now believe is the first American English usage guide, Seth T. Hurd’s Grammatical Corrector or Vocabulary of the Common Errors of Speech (1847), I found an entry on eenamost:

And eenamost is in the OED. What is more, the first illustration of the word suggests that it is a Kentish dialect word: “1735–6 S. Pegge Alphabet of Kenticisms (1876) , E’en a’most, almost”. Elizabeth and her family lived in Godmersham, Kent, and in this letter to her aunt Cassandra, she mimicks the language of “Poor Will Amos”, a local 65-year-old Godmersham villager (see the index in Deirdre Le Faye’s edition of the letters).

The OED labels the word as “Eng. and U.S. dial.”. Hurd, however, labels it as “a gross corruption”. Perhaps there are more dialect words in his book which he labels like that. Searching for the word with Google’s Ngram Viewer (set to American English) suggests that the word was particularly frequent in the 1840s, and this is why it may have been adopted in his list of “common errors”:

Perhaps usage dropped as a result of Hurd’s book, but we need to know a lot more about its distribution and popularity. Has anyone ever come across references to it? Usage also dropped in British English after the 1840s, but it never surfaced again as much as it did in American English.