If the 30-06 is too much, the .338 win mag isn't going to help. More powder, more bullet, more velocity equals more recoil. If you handload, you can make the .30-06 do whatever you want in terms of lowering the recoil. 125 NBTs over a light load can be handled by just about anyone.

Plenty are using the 6.5 Creedmoor for elk. I'm not sure a little more isn't better. The .270 is probably still the best lighter recoil option for bigger ungulates. Think well constructed 150's. The distance you are willing to shoot and weight of rifle you are willing to carry play a large role in cartridge selection when recoil is a factor.

Recoil pads and brakes (if you can handle the noise) can do a lot for recoil.

There are better options than Leupold.