Sony is putting out the soundtrack to the movie, but Amazon hasn't got it yet. Probably has poor version of the folk song from California.

"El Sombrero Blanco" was a folk song recorded from the Villa Family and others of California in 1904 by Lummis; sheet music is in the several volume "Los Californios" transcribed by V. M. Gray. Available from San Diego Friends of Old Time Music. See: Los Californios
A cd of the real Californios music, including "El Sombrero Blanco," is available for $15.
Order form: http://order.loscalifornios.com

An old Mexican song with the title "El Sombrero", not the one you are looking for:
El sombrero que me pongo,
lo traje de San Martin;
a veces que me lo pongo
vengo a escuchar el violin
y me pongo mi jorongo.
(And probably dozens of others)

An earlier "The Mask of Zorro" with Tyrone Power and Linda Darnell (1940) was titled "El Sombrero Blanco," and also used the song.