I have 2 338s and one 300 RUM. I think the 338 is a better big game hunter than the 300 RUM. I use the 300 RUM for target and some long range deer hunting (500 yds+).
Of course there is a lot of overlap in the 338 and the 300 RUM. If I was to be thinking of another big rifle I'd go larger, 416 Remington magnum. My 416 does kick with the 400 grainers, but it is in a different class than the 300 RUM or the 338 mag. I've killed lots of game with the 416 and never needed more than one shot to get the game on the ground immediately. I've never shot an elk with the 416, but its killed lots of stuff: ground hogs, moose, grizzly bear, cape buffalo, kudu, etc.....I'm sure if elk is your target it would do great. With the 350 grain slugs, my 416 shoots as flat as a '06.
Just my 2 cents.