Maj. William Child, M.D." A History of the Fifth Regiment, New Hampshire Volunteers, in the American Civil War, 1861-1865" (1893):

“[On March 17, 1862] marched ten miles, forded streams several times. The colonel ordered no delay at these fords and would enliven the passage through the water by a tune from the band, which tune was familiarly known as ‘One-eyed Riley.’ It was a lively and inspiring air; the men must keep time to the music, hence they crossed without delay.”

Presumably Winner's tune, or something quite close to it.