If I had to put together a list of the guns every gun owner should have, it would be something like this:
Centerfire pistol of caliber 9mm/.38 special or greater
12 or 20 gauge shotgun
.22 rifle
Centerfire rifle capable of taking deer-sized game
You have the first and third categories taken care of. I'd say a shotgun should be your next purchase, and find a range that allows you to shoot it (what kind of range allows .308 but not shotguns?). A shotgun would also probably be the least expensive--a NIB Maverick 88 is less than $200, and a new 870 only runs about $300.
If you really want to get a centerfire rifle, I'd go with the .223. I know it doesn't satisfy my "capable of taking deer-sized game" criterion, but it's the best choice if all you are doing is punching paper--ammo is cheaper than .308 (I'm assuming you don't reload), and it doesn't burn out barrels like .243 does.