The national tour of “Hello, Dolly!” just opened the second leg of its 10-month multicity run at Chicago’s Oriental Theatre. But the frothy, feel-good musical has turned into a whodunit.

When “Dolly” premiered in 1964, its irresistible title song cemented composer-lyricist Jerry Herman as the heir apparent to Irving Berlin and Frank Loesser—king of musical-theater optimism. Mr. Herman went on to compose two other megahits, “Mame” and “La Cage aux Folles.” But “Dolly” put him on the map.

Which...