The show toured the US from October 15, 1959 until it landed on Broadway in January of 1961. The following theatres, with no dates given, have been traced via Playbills. Because no dates are given, and the first instance of Jules Munshin joining the company is June 6, 1960. For the moment, that is our litmus for when the show was in these venues.
The Palace Theatre scene (consisting of "You Haven't Lived Until You've Played the Palace" and "Somewhere There's a Little Bluebird") is largely a pastiche/parody of Betty Hutton (in dialogue and dress) and Judy Garland (in song and physicality).
The first act ended with a long scene entitled "Carol's Musical Theatre." It was a pastiche/parody of the current style of Broadway musical. It was broken into three sections of two songs each.