Rhymes for Fingers and Flannelboards, by Louise Binder Scott and J. J. Thompson. Illustrated by Jean Flowers. McGraw-Hill Inc., 1960.
Can't help but be reminded of the Saul Bass design for The Man With the Golden Arm, 1955. Note: if you are editing a cover for children's book in 1960, drawing inspiration from a very famous poster design for a movie about heroin addiction is probably not the first association you're going for. It's still a great book though.