The lyrics and music in the "Girl Scout Pocket Songbook," 1956, Joe Offer, above, are identical with those in the older "The Ditty Bag," 1946, Janet E. Tobitt, New York. The lyrics in the latter are marked to be sung "With a riding rhythm," p. 50). It has the melodic 'contours' noted by Cruiser.
I haven't seen the Ginn and Co. school song book, so can't say if the melody is the same.

The notes vary somewhat from those in Ohrlin, until line four, where the difference is marked. "With foot in the stirrup" descends from a high note and "I gallop away" is low. In Ohrlin, "With my foot in the stirrup" notes remain high with a hold on stirrup, and "I gallop for aye" is moderately low. The effect is quite different.