It is important to determine what you mean by wanting to shoot big game at long range. 98% of shots by long range hunters are less than 1/2 mile, 880 yards. Many guys don't have the ability to shoot even that far.
The 338 Win Mag is certainly capable out to 1/2 mile if you have an accurate gun and you select higher BC bullets. At 700-750 yards you would still have over 1800 fps, to ensure bullet expansion, with a good 225 grain bullet. With a softer target style bullet you don't even need 1800 fps to get expansion so the distance could increase further.
Not trying to talk you out of buying a new gun
, but it sounds like you already have one now that may work with some tweaking!
If you are thinking of going further then a new cartridge and gun may be in order!