I kind of like the idea of just upgrading your scope.
The is a big difference however, between, say, a .30-06 and a .300 RUM or .300 Weatherby. The latter two produce a lot more downrange energy. The .300 RUM is still pushing almost 2800 Ft. Lbs at 300 yards.
Not sure you need that much power for what you're going after. I've taken too large bull elk with my old .30-06 Browning BAR. I got too lustful and ended up getting a .300 RUM in a Rem 700 BDL.
Good luck