I suspect that 'Sweeney's Men' learned the song from a Pete Seeger or Peggy Seeger record; they both recorded arrangements of it, having in their turn got the song from a recording of Troy Cambron, made in 1940 as part of a documentation project at the FSA migrant work camps in central California.

See Voices from the Dust Bowl:

Johnson-Jinkson

See also Remembering the Old Songs:   Johnson-Jinkson

'Matthews Southern Comfort' recorded a further arrangement of the same version a little later; they could have learned it from the Sweeneys or the Seegers.