Just came across this old thread. I remember "I'll Take You Home Again Kathleen" in one of the movies, and I remember some Stan Jones songs in some of John Ford's westerns -- "Chuckawalla Swing," and another one, "Wagons west are rolling, out where winds are blowing," Another I can almost remember. Of course Stan Jones wrote Ghost Riders in the Sky, but I don't remember Stan Jones ever singing with the Sons of the Pioneers.
Ken Curtis, who was one of the Pioneers for a time, sang Wild Colonial Boy in the Quiet Man.
I went to those movies to hear the Sons of the Pioneers and to ogle Ben Johnson. That was a couple of hundred years ago, of course. A reviewer of the time wrote that the S of the P, sang "quasi cowboy songs, and contributed to the ennui." I had to consult a dictionary before I could disagree with the reviewer.