There are any number of cheap pocket sets with discs for pieces that fold up and are very pocketable. Search on amazon for pocket chess set or, if you happen to live near a decent games shop, go there. They won't cost you much.
flat, pocketable chess set to take games along to study...

Not sure how thin it needs to be if the ChessMate was too thick, but you could check out this one:
https://www.uscfsales.com/chess-sets/travel-analysis-sets/checkbook-magnetic-travel-chess-set.html
Looking for a flat, pocketable chess set to take games along to study. I did try a ChessMate wallet folding magnetic set. It was nice, but too thick. Any ideas?
I do not own a smart phone (which would be ideal - an app where you can set up a board to an existing game).