Far, far more likely to be a .30-06, .308, .270, 7mm, .300 mag than a .338 magnum. Anyway, any of those calibers would be more than sufficient for a cougar (so would the .40 caliber pistol but a typical shooter would be far more likely to hit the target with a rifle than a pistol).
A couple of my friends in Oregon who hunt elk use the .338 Lapua.
Some others I know use a 7mm Rem Mag. Me, personally? I’m a .30-06 guy (though I’ve never hunted elk).