It is worth noting that Ewen Hail's contemporary, the widely popular Vernon Dalhart, also sang a version of the Cowboy's Lament, and his tune differs from both the standard one and Hail's.

I can best describe it by saying that it's the tune I remember to:

I am a vaquero, I wear a sombrero,
...(?)... my pony is Taw,   etc

It is much more cheerful than the other Streets of Laredo tunes. Dalhart can be heard singing it on YouTube:

https://www.youtube.com/watch?v=yc97GED-HB0

Bob