I'm a big fan of using the same model for everything. If you can get a 375 in the same model as your 338, that would be the bee's knees. Muscle memory and auto pilot are pretty handy tools.
Bfly
I am also a big believer in this. Although they are not the same make, 7 of my hunting and practice rifles (.223, .22-250, .257 Ackley, 7 mm RM, .308 Win, .300 Wby, and .375 RUM) are scoped, bolt actions of approximately the same weight, have the stock dimensions, and the same safety location and operation.
When I was preparing for my first Cape Buffalo hunt in 2005 I had a Rem 700 Classic in .375 H&H, but it's stock was not the same as my other rifles, and it wasn't exactly what I wanted, so I traded it and bought a Stainless Rem 700 in .375 RUM. I immediately re-stocked it in laminated wood from Richards that I shaped and checkered the same as my other rifles. I then developed moa or better loads for it with several different bullets (300 grain TSX @ 2840 fps, 300 grain Partitions @ 2909 fps, and 270 grain TSX @ 3030 fps). I put a 2-7x Leupold scope on it, and to control its recoil, I had a KDF muzzle brake installed on it and I put a mechanical recoil reducer in the stock. It's felt recoil is not much more than my 7 mm RM.
I have used it on two South African and one Zimbabwe hunts, with shots from 30 to 348 yards on a variety of animals from Steenbok to Buffalo, and I couldn't be happier with it's performance.
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