We used to know it as the soft cap, one each.<< O.D. in color
When I was in the Army they issued us fatigue caps that were soft and shapeless. If you tried to fall out wearing one you would get chewed out and written up for being out of Uniform. I pointed out that it was issue so how could we be out of Uniform and was told to shut up and do what I was told. We had to buy the ones we actually wore at the PX, they were blocked and looked a lot nicer than the issue ones , very similar to your picture but not camo. At any rate I do think they are much better than a beret.
I like the choo, choo, hat. Is that the Marine type?