"When McGuiness Gets A Job," which is the vaudeville original of "Last Winter Was A Hard One" dates from 1880 when Italian immigrants were challenging the Irish in New York with cutthroat wage competition for manual labor jobs. For a few reasons, "Dollar A Day" would seem to come from the same time and background.

I have not run across "Dollar A Day" in looking through 19th century American songsters but that's one place Mrs. Lev might like to look.

All the best to you and yours,
Dan