Unless you have other plans later for the 338 I would stick with a 30 cal. Anything from the .308 Win, 30-06 , 300 wm, 300 PRC, or my favorite 300 Wby Mag. The logical is the 300 WM as ammo is easily available.
You can do it all with a 7mm also. The 338 isn't large enough for dangerous game...