I think it's somewhat misleading to say it was a "poem"; Shel actually was quite the songwriter in earlier days, and I'm pretty sure ...Sue was written as a song, not a poem. He also wrote Drop Kick Me Jesus (Through The Goal Posts of Life).
Well, no, it's singspiel, as it were, but according to Wikipedia Silverstein wrote the chords too, presumably intending the both of them to work together.