If we get practical,you have a .308 and a 30-06.Fair to say you can hunt anything in the lower 48 just fine with those.
But,who wants to be stuck with practical?I think the .338 is a great cartridge(and so is the 338-06).
If you choose to,look at the 165-168 30 cal bullet BC's.and the 250 is grain .338 bullets.I believe you can find near identical BC's.I also think your load book will show near identical velocities.You can have a.308 and a .338 as a pair.
What is learned from the .308 will work real close with the .338.
Something to consider,I like a push and roar of shooting a big gun,but everyone has limits.I tried a .416 Rem Magnum.Its not fun for me,I do not want one.I enjoy my .375 Taylor A 44 magnum handgun is great,but full power 454 Casull is too much,for me.I built my .257 AI for a good reason.Less is more,sometimes.
You have to decide if you enjoy shooting it.Important,do you start to anticipate recoil and get a little squinty?One way to notice a decrease in your shooting skill is to shoot a gun that is uncomfortable to shoot.You can then recognise you are flinching and dsicipline yourself to recover but that might happen after you miss your 150 yd bull.