Since I don't know the Ry Cooder recording, I cannot say for sure that it is exactly the same tune, but Fiddlin' Sam Long recorded Seneca Square Dance in either January or February of 1926. He is generally considered to be the first Ozark musician to record. The song is available on Echoes Of The Ozarks, Volume One, County 3506, 1995. The album is an excellent anthology of mostly Arkansas musicians recorded 1926-1933.

Another good version, derived from Long's, is by Geoff Seitz of St. Louis on his The Good Old Days Are Here CD, Oceana 003. It is not as widely available, but you can get it through Elderly.