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There are a number of alternatives to a fanny pack
I often use one of those "daytimer" holster cases. Holds a G19 and a spare mag, looks innocent enough. Have a company name silk-screened on it for effect.
There is a variation on this that is a real functional daytimer, with rings and a calendar, note pages etc. But half of it is a zipped compartment for "other stuff". It would hold a Ruger LCP or similar and a half dozen mags.
I also have a Keltec P32 that fits into a manicure case, with a spare mag. It's the size of a wallet, and has some convention title screened onto it. I bet very few people would see it as a gun case.
My wife keeps her Beretta Tomcat in a tan leather GPS case. Lots of camera and personal electronics cases work very well.
Even CD cases, though those are a bit of an anachronism. I stored a full-size 1911 in one for years.

All these are everyday containers that have a zipper closure, and are easily re-purposed.

Another alternative: NAA Black Widow on a neck lanyard. Popular with runners and cyclists