Because the USAC / USAAF was part of the US Army, officers and enlisted men were issued the same basic uniforms and headwear. Dued to the nature of their work, the flyboys were allowed (by regulation) to remove the stiffener and grommet from their service caps in order to facilitate the use of a headset over the cap. This practice resulted in the famous "crusher" look and this is how USAAF caps became known as "crusher caps" or "50 missions caps". A lot of bomber and ferry pilots wore such caps instead of a flight helmet.
Some uniform manufacturers started to make pre-crushed caps exclusively for Air Force issue. These caps had a thinner and more flexible leather visor and no stiffener. Some well known manufacturers are Bancroft (the flighter), Dobbs Fifth Avenue, Lewis (fly weighter), etc.
The overseas cap or garrison cap used by the USAAF is easily identified by the piping : black & gold for officers and warrant officers while the enlisted men had an orange & blue piping.
Although the service caps were used to fly, the USAAF develloped caps intended for active combat wear, the B-1 & B-2 caps. The B-1 is a summer version made of canvas material with a long bill while the B-2 cap is the winter version. The B-2 cap had a shearling band all around the head which could be flipped down if needed to protect from the cold.
Flight caps