Here's a modern recording of the song: Hunting the Buffalo - Big Chimney Barn Dance

Here's a recording from Ramona Jones, wife of Grandpa Jones, from her album Back Porch Fiddlin' Vol. 1 (Happy Valley Records PRP-4387, 1973). She titled the song "Gonna Go Hunting For The Buffalo," and wrote the following about the tune:

Jimmy Driftwood of Timbo, Arkansas played this tune for me and it had such a haunting melody I couldn't get it out of my mind. He was kind enough to teach it to me and tell me the history behind it. His two uncles spent years on the Cherokee Indian Reservation, and when they returned, they were playing this and other old Indian melodies on the fiddles.

TuneArch similarly attributes Jimmie (or Jimmy) Driftwood. You can hear his recording here

I can certainly hear the similarities to Cherokee Shuffle.