There is a listing in American Memory for a song, anonymous, popular in the 1840s (1844), called "The Ole Grey Goose." It was printed in Philadelphia, A. Fiot, words and music.
The reference is in "Greatest Hits, 1820-1860." Greatest Hits

They have a copy with that title, by Chas. Reps (arranger), published by C. O. Christman, New York, 1844. It is a minstrel tune (Boston Minstrels) and not related to "Go tell Aunt Rhody."

The sheet music is notable for an engraving showing musicians with their instruments, including accordion and flute.