Yes, the song--at least the tune--goes back to at least the 30s. I once found a recording of "Shorty George", an early version of HWaFoM.
So, perhaps the tune was very familiar to cowboys. Kinda like Ennis humming "The Streets of Laredo", before coming on the bear.
A little aside: an early version of the screenplay includes directions for various old Western songs to be placed in particular scenes. The titles often cleverly related to the action, one way or another. Of course, I can't think of one right now... I did take notes once. They are all different from what ended up in the film--like "It's So Easy", D-I-V-O-R-C-E", etc.
In Tucson, I asked Larry about it. He said that, while there were musicians in his family, the music was Diana's contribution!