I sing a different tune to Streets of Laredo, dating back to when I first heard the Supertone recording of Ewen Hail (The Cowboy Minstrel), recorded in New York March 31, 1927, issued on Brunswick and Vocalion as well as on the record I heard, Supertone S2043.
Hail did not play an instrument; he was accompanied by Bert Hirsch and Carson Robison on fiddle and guitar, respectively. Hail also sang popular material for the Vocalion 15000 series catalog, but in this recording he sounds plenty authentic, comparable to Carl T. Sprague and Jules Allen, his best-known cowboy singing contemporaries of the 1920s. His mournful (though major-key) version is irresistible.
Luckily you can hear it on youtube:
(help, someone, can you make this a blue clicky? I'm a duffer at it)
It seems likely this recording is the alternate tune discussed above. I don't know where Hail's version originated (he may be its composer), but IMO it rates a new life among traditional singers. I'm tickled to recommend it—give it a listen and see what you think.