I worked for a crop-protection products company for many years as a tech service rep and gimme caps were, as mentioned above, a form if currency. Polo shirts, winter vests, and quality winter coats were premiums above caps and all went to good & faithful customers. All had colorful logos of our products mostly. They were popular as most of our trade names had a sort of military theme like "Scepter" or "Pursuit" and were popular products in their day.
Purchasing them, to give away, always came out of your own budget but woe be to the rep that didn't have boxes of "business-builders" in the truck ready for dispersal.
I had a fine, large, collection of never-worn ag-related logo gimme caps until it got downright silly - and I needed the room for firearm storage. I divested thousands of them, much to my wife's relief, donating them to kids clubs and such.
I heard "gimme-cap" a lot when I was in school in North Carolina in the early 80s and my wife has always called them that - she's from Tennessee. Seems most everyone in rural Texas knows what a gimme-cap is.