I suppose you could say that any song in a musical remake of a film was inspired by the original film.

For instance, the movie "A Star is Born" (1937, starring Janet Gaynor and Fredric March) was not a musical, but there are two film remakes of the same title that are musicals (1954, starring Judy Garland and James Mason; and 1976, with Barbra Streisand and Kris Kristofferson). While it's a stretch to claim that the Streisand-sung love song "Evergreen" was inspired by the 1937 movie, certainly Judy Garland's "Born in a Trunk" medley/sequence describes the years of hard work required to become an overnight success, which is what the female lead in all 3 films is.

Then there's the Broadway musical "Spamalot", which is loosely based on the movie "Monty Python and the Holy Grail" but contains several new songs that were not in the film. (It also lifts "Always Look on the Bright Side of Life" from the Python film "Life of Brian".)