I have just been watching the recently released DVD of this movie, which stars John Wayne, from the novel of the same name written by E.K.Gann. The story is closely based on a true incident in which the author participated. This occurred in 1942 when the US Transport Command had just been formed and was using mostly civilian pilots seconded from many US airlines. Gann himself came from American Airlines. These pilots wore whatever they pleased, though many of them continued to wear their airline caps.
This situation is also pictured in the movie, but for some reason all the C-47 planes in the film have polished metal finishes and carry the titles 'United States Air Force'. I believe that this title came into effect in 1948. Before that it was the US Army Air Corps and then the US Army Air Force. Since Ernie Gann wrote the screenplay himself I can't imagine that this is just a simple mistake - though with Hollywood anything is possible. If, on the other hand, the story was updated to the post-war era, why do they still have civilian pilots flying USAF Transport Command aircraft?
As Yul Brynner said in the movie 'The King and I' - " 'tis a puzzlement".