In the hymn or white(?) gospel song, the line is "Every rung goes higher and higher," whereas in two 'spiritual' versions given here, 'rung' has been morphed into 'round.'
No idea about the age or true character of the hymn; the versions I have found, called 'traditional American,' have had material changed or added.
I would like to find the Methodist hymn to which Allen (1867) referred.