At a singing "jam" session during an Augusta Heritage Vocal Week (in West Virginia) many years ago now, I remember a group of us standing around one evening just singing, vocally riffing & improvising harmonies, for 15-20 minutes straight... (it was sublime!)

IF I remember rightly, we discovered that evening that "Mama's Little Babies (Love Shortnin' Bread)" went together very well with "The Atchison, Topeka & the Santa Fe" (or at least the chorus -- don't know about the verses).

Then there are rounds... "Ah Poor Bird" and "Rose" go together beautifully, and "Zum Gali", "Hey Ho, Nobody Home" and (the chorus of) "God Rest Ye Merry Gentlemen" also go together very nicely. If you have a copy of "Rounds Galore I" (or can get hold of one from a friend/library/online bookstore), it lists "Come With Me" and "Twinkle Twinkle Little Star" as quodlibets, "Four-Way Gospel" as a "Trad" Quodlibet, and also "Windmills" and "Jubilate Deo" as partner rounds.

Hope this helps...

BTW (tho I know It's not what you're looking for, Genie), another Music Man song with two parts that get sung together: "Good Night Ladies", and "Pick a Little Talk a Little"...

And of course, almost all of Lou & Peter Berryman's songs consist of two distinct melodies -- first sung separately, then simultaneously.