I keep a USGI M-16 cleaning kit on my ruck. That's good enough for every gun I own. These are super light and can be stored on the outside of the bag. In a true SHTF scenario there will be plenty of things to scavenge so I don't specifically pack a big/full cleaning kit since it's really easy to clean guns with scavenged materials.
The one thing I do like having is a rigid rod that can be used to clear obstructions in a hurry and that's why this kit trumps the pull through kits like the Otis or BoreSnake.
ETA-- For SHTF situations I'd add a ruptured cartridge extraction tool and/or a set of curved hemostats. It's possible to knock a ruptured case out with a rod, but it's a lot more time consuming. When you're doing a lot of rapid fire or machine gunning, ruptured cases happen. If it takes too long to clear.... that might be all she wrote.
.....and just re-read through here and noticed you said handgun. My bad.
The military M-12 holsters and several models of European guns like the Toks and Maks sometimes come with little cleaning rods as part of their holsters. Having something like that enables you to rapidly clear obstructions and get back in the fight. At minimum, I'd have a stick or some type of rod like that.