Here in the midlands, a "gimme" is anything that's free.

Extends naturally to the "gimme cap," usually from the local co-op.

Same term is used (e.g.) on the golf course where someone thinks their putt is so close they ought to be able to "count it in" without actually playing it. "That's a gimme, ain't it?" (Not done by honest golfers.)

The feed caps go back to long before nylon mesh was invented, and were usually cotton, whereas the old style ball caps were commonly felt, or at least a "felted" or "brushed finish" cloth.

The really old feed store "gimme caps" were often a "pleated" shape, more like those often worn by railroad workers - (make sense?)

Until recently (say 1945) the feedstore caps might have a stitched brim of several layers of cloth, but almost never had the cardboard "stiffener" in the bill like ball caps. (You really needed to be able to stuff them in your hip pocket (or inside your bib) on occasion.)

John