.338 WinMag is NOT a competitor to the .338 Lapua, in my opinon. I had one. Not what I'd call a LR hammer. To me, it seemed kind of like the goofy unbalanced cartridge... Like a 600lb guy with toothpick-sized legs. Too top-heavy with not enough boiler room to push that big bullet fast and hard enough.
Could have just been my experience...And I don't mean to offend anyone who likes the caliber. It was just my assessment of it.
If I were in your shoes, I'd build a .338 RUM or a .338 Edge.
Although, a well-built and accurate 7mm STW is a force to be reckoned with for LR shooting.
If you are wanting to rebarrel your new rifle, I suggest giving the STW caliber a chance if you handload, and buy a new aftermarket heavy barrel and have it installed.
If you just want change, the .300 Ackley Magnum can use 7mm STW brass to shape and fire-form. You can also use .300 Wby, or .300 H&H brass. It's one hell of a LR hammer. My gunsmith has a couple. He really loves them. I am building one soon, too, as soon as I get a donor magnum 700 action.