The New Song Fest (1955-1958) does not have the paratroopers' version, but it does have the skiers' and the rockclimbers'. Each of them sticks to its subject. The rockclimbers' one does not have the stanza quoted. It begins

"Will it go around the chockstone?" called the belayer, looking up.
Our hero feebly answered "Yes," and slowly inched on up.
He was trying to drive a piton when his foothold crumbled out,
And he ain't gonna climb no more.

    Gory, gory, what a hell of a way to die! (3x)
    Oh, he ain't gonna climb no more.

(I have emended "chalkstone" to "chockstone" & corrected the punctuation.)