O.k. Malcom, this is the tune, to the best of my ability. Bear in mind I'm but a v. clumsy ABCist.
T:The Rose In June M:4/4
% ABC2Win Version 2.1 6/13/2001
This tune is that of the repetitious ballad verses. The hymn is distinct. Its contrasting rythm and feel breaks the repetition of the ballad tune, and it seems to me this is what makes the song really interesting. So far as I can ascertain, the hymn is "I am Thine," Salvation Army 739; the words and tune are not available on line. But it bears a close family resemblance to "Revive Us Again," a standard Protestant hymn. Which has exactly the same tune as Hallelujah, I'm a Bum.
Killen's hymn tune is a bit different, and I have no idea whether it was he who changed it, or the Salvation Army, or the person he collected it from. I'm not sure how many people would spot the difference, and unless someone is really interested in the hymn tune, I will let it go.