I have 100 rounds of new Federal 338 brass if you are interested. Bought it and then never built the gun.I have a 338 Federal and a 338 Win. Mag. and both shoot well, with 180 Noslers for Deer and 225 Barnes for Elk. The Mag will reach out farther but with more boom. Thinking about something in between maybe a 338 X 284
My friends lefty Mark 5 .340 was Magna Ported. That tamed the recoil quite a bit.The .340's a good one, I ran amuck with one for close to 2 decades. Shot out two Schneider barrels. One on a 700 and one on a P64/70.
Incredibly accurate, used it on a ton load of chucks, yotes, porky's and deer, lope, elk and black bear.
I enjoyed the run but in the end I decided I can do the exact same thing (kill on demand) via a round with a ton load less recoil.
Bottom line, you have a 06, a 300 and a 375 life doesn't get any better than that. I'll say it again, move on from the 338. Use the funds and the time to become more intimate with the guns you have and get to hunting.
Life's short, hunt hard!