You will have more options for bullets with a high BC with the .338.
Find the LONGEST bullet in .338 you can find with a boat tail. The boat tail is extremely important. I just looked up the stats for the Lapua Scenar bullets and DAMN. The .338 Scenar has a Ballistic coefficient of .785. That's extremely impressive.
I wanna say i have seen some close to .900 in the 338 cal but cant remember the brand. I guess i should say i want a bullet thats less than $2 dollars a pop lol i know some lath turned bullets get freaking pricey