.30-06 is a good cartridge, but it isn't going to be a significant improvement over what the OP has in .270 and .308 Win when it comes to elk hunting. All three cartridges work really well for elk.
What modern rifle uses a "magnum length" action? They're all SA or LA, the M70, M700, Savage 110, and Ruger 77 use the same action for the .30-06 is the same length as the .375 H&H and other long magnum cartridges. Bolt stops and magazines are different and some other differences, but the action is the same footprint. .300 Win Mag and 7mm Rem Mag were designed to work in the same length magazine as the .30-06 cartridge, as well as alot of the other magnum cartridges. Can't help you with your belt phobia, get suspenders!
That's the beauty of most hunting rifles and cartridges suitable for hunting. They most commonly come in one of three bolt faces, .384, .473, and .532. This covers the majority of the rim diameters found on current cartridges. This covers .223, .30-06/308, and .375 H&H/.300 Win rim diameters and a plethora of others.