You don't have to carry in a fanny pack in the summer. Get yourself a compact handgun, an inside the waistband holster, kydex won't sweat through. Buy a belt meant to carry a gun on and then dress yourself. You'll probably want to wear a T-shirt to keep a kydex holster off your skin, and a light cover shirt untucked.
I am going to stress buying quality and doing research before you end up with a bunch of Uncle Mikes nylon holsters in a drawer collecting lint. Expect to spend $50 to $150 getting a good holster and the same on a good belt. Look around, nothing says GOOBER like a fanny pack, and fumbling around with a zipper and a gun is a good way to be unprepared.
Alternately an airlight S&W J frame in a pocket holster makes for a pretty low profile carry weapon. Just make sure the pants are loose enough through the thighs to get your fist in there. I grew up in Il. and I know what the humidity is all about so I understand your concerns on concealment.
You aren't going to be able to carry off the tri athelete in training look as often as the guy going to work or the store look. The gals at the C-Store will instantly know you aren't a felon when you produce your check card and leave with the 44oz soda and a newspaper. When carrying a pistol is legal well ...it's legal and besides if it is a conceal license they won't see it anyway.
Once you get past the novelty of it all you find that nobody is paying any attention to you anyway, they never did before and they aren't likely to start. ( teenagers have to hang christmas ornaments from their eyelids just to get someones attention)
When it's all said and done the Cook County crowd will nix this in all likelyhood. Like the early settlers of this nation you may have to move to escape oppression. Heck I did it, the air is just fine up here, and the fish taste better too!