It's slightly ironic that a history of Pan Am should be called 'Come Fly With Me' as the song was originally written to boost the fortunes of TWA. Although I haven't seen the documentary yet, I suspect they used the song.
TWA were in trouble in the mid 50's. The long overdue Lockheed Constellation was sending them under and the owner, Howard Hughes was suffereing assualts from Juan Trippe on several fronts, including a congressional inquiry which was probably instigated by Trippe. Siantra was a friend of Hughes at the time and although denied in his autobiography, the album 'Come Fly With Me' was a thinly disguised advert for TWA, you only have to look at the cover and see the red and white Constellations to realise that it was true
Later, Hughes and Siantra were to have a major falling out. Hughes objected to Sinatra's links to the Mafia, and Sinatra marrying Hughes' ex girlfriend, the actress Ava Gardiner probably didn't help either!