Hmmmm, what comes between .30 cal and .338? Why its the greatest orphan of all times, the 8MM Remington Magnum. With 77 grains of IMR 4350 and an 8mm 170 grain Hornady SST, it'll squirt out the barrel at about 3100 fps. It will test your recoil resistance! I've killed Mulies with one shot each of the last two seasons (my favorite gun-expression "Bang Flop"). Don't hit anything that you might want to eat "cause it will hit with shock and awe" at closer ranges. I have 200-grain Swifts on standby as I wait for my Wyoming Elk Points to build up.
My Rem 700 has a new Bartlein 26" ss barrel and will shoot tight at the range. The full length, long-action, aluminum-reinforced Hogue stock does a pretty good job of shock absorbance but as I say I never feel or hear anything with a antlers in the sights when I squeeze the trigger,
Oh yeah, and if you want a big bullet with less recoil, I've got another Rem 700 in .270 (shot-out barrel) being bored out to 8MM-06 by Danny Pederson at Classic Barrel and Gunworks out of Prescott Arizona. (I've also got another Rem 700 with Danny who is going to rebore (if he can) the barrel on another .270 out to a 7mm-06).
One thing that helped keep 8mm down was the lack of bullet choices in the past, that is not the case now.