If you already have a .30-06, you're covered for any big game in N. America, although one might want a bit more "insurance" on brown bears.

I live in a state where the minimum is also the .243. In-between it and the .30-06 are quite a few great calibers, among them: .25-06, .257 Roberts, .260 Rem, 7mm-08 Rem, .270 Win, .280 Rem, and the .308 Win, as well as a smattering of 7mm Magnums. None will do you a whole lot better on the large end (moose, caribou, etc.) than will your .30-06. On the other hand, a good number won't do you a whole lot less on the small end (deer, varmints, furbearers), than will your .30-06.

My experience has been with the .243 Win, which has proven itself more than adequate on deer and antelope, and with proper bullet selection, not overly destructive on coyotes and bobcats, and perfect for prairie dogs. I would expect the 6mm Remington to be similarly versatile.

Really, unless you drop down to a shorter case length and caliber(7mm-08, .260, or .243), you haven't really left the class your .30-06 is currently in. And, of those three cartridges remaining, my choice would be the .243 hands-down.