I went through this myself a few years ago. That was when I learned that the .338 Win Mag was not a hotrod cartridge (as Mudrunner said) like I thought it was at the time. It still isn't, but there are better bullets available now, which can dramatically improve its performance. If you are determined to have a Winchester Model 70 in .338 Win Mag for use at long range, Cutting Edge bullets are worth a long look. They are expensive, but they can give you everything that can be had within .338 Win Mag case capacity limitations.
With available bullets and powders, the .300 Win Mag will far outperform its .338 counterpart at long range. It is an easy and inexpensive solution compared to taking the step up to the larger cased .338's. I have a .300 Win Mag in process, myself.
If you want a large cased .338 with relatively easy factory ammo availability, the .338 Lapua is the way to go, IMO. I have consistently seen ammo on the shelf at big box type stores for several years. Off-the-shelf rifles chambered for the Lapua are also not hard to find. The issue is cost. There is nothing inexpensive about shooting the Lapua. Available, yes. Expensive, also yes.