In such a situation the only caliber worth while is one you can get ammo for.
Here’s my assessment. Chances are you may not remain stationary so you may have to pack everything. Stockpiles of ammo, food, and clothing will not work. You will not be driving long. Pack animals, maybe, but you will eventually become a target.
There are great air rifles on the market, you should possess one. You should have a mold for making pellets or bullets. Lead is plentiful in car batteries. A compound bow will be indispensible and often arrows may be reused or made more easily than firearm ammo. You can quietly feed yourself with an air weapon and/or bow.
As far as caliber, what ever is most plentiful.
Before jumping to that conclusion I'd suggest attempting to make decent arrows from recycled or raw materials. I've not tried it myself, but I suspect it's not easy.
Sage advice.
Lead is very common in car batteries, but you really do not want to melt them down, there are a lot of nasties in there that will kill you pretty quickly. I make sinkers and all manner of lead fishing stuff, and i will not touch car batteries for this purpose. Find a different source!
I agree. Just got a .25 caliber pellet rifle - is very quiet, with subsonic muzzle velocities and quiet action. Quiet is good. Pellets are plentiful and cheap. You pump it up with a souped-up bicycle pump.