Roger, you're exactly right about the gimmee cap, though I'd prefer to spell it gimme cap. Gimme is a slurred pronunciation of "give me" (or was that obvious?) referring to their giveaway nature. Google gave me 13 hits on "gimmee cap" (in quotes); 502 hits on "gimme cap"; and 49 hits on "gimmie cap".

"Gimme cap" is even listed in The American Heritage® Dictionary of the English Language: Fourth Edition, 2000, although it doesn't give much explanation.

I'd be interested in knowing how many terms there are for this familiar kind of cap. The first term I heard was "feed cap" because they frequently advertised a brand of animal feed and were given away to farmers by the feed dealers. I have a genuine feed cap advertising Supersweet Feeds, from Chuck's Feed & Grain, Peterson, MN.

Before that, they were probably called "baseball caps" because they have the same general shape as a baseball cap, but a purist would say they're a completely different thing. Baseball caps are made of cotton fabric; feed caps are nylon mesh. Baseball caps come in different sizes; feed caps are "one size fits all" by virtue of an adjustable plastic strap at the back.