While we don't exactly know what your definition of 'big game' is I'll enforce what Float Pilot says.
Both the 30-06 and the 338 Winchester Mag. are prolific calibers, easily found in most corners of the land.
The 30-06 will handle most game up to and including the ever elusive Elk creature, and preform this stunt out to 500y or so, shooter able. lol
The 30 caliber Magnums offer what the 30-06 does but with more energy available, and for a longer duration, it also offers faster and flatter flight paths. The 300 Winchester Mag. is, as the 30-06 easily found in most parts of the land.
The 338 Win.Mag. as Float Pilot says, will handle what you ask of it to 400y, give or take, and again, shooter able!
The 338 Remington Ultra Mag is simply phenomenal in all aspects as are ALL the Ultra Mag cartridges, however the Ultra Mag line IS expensive to use and is not as readily available as the older magnum calibers such as the 338 Winchester Mag.
As for the pipsqueak calibers and 'big game'... not the best idea! IMHO Without offering any idealistic beliefs, the 6mm's, while extremely accurate and flat(er) shooting than the 30 and 33 calibers do not posses the punch needed for even deer at extended ranges, much less the Elk or Caribou creature.
The old 'Military' calibers, excluding the 30-06, 300 Win.Mag. and the 308, while cheap to use fall to somewhat inaccurate performance if military ammo and or firearms are utilized.
Hornady is now loading the 9.3X62mm and the 9.3X74R which would qualify for your over 30 caliber desires but are expensive to shoot.
The Weatherby calibers offer a bit more than the standard magnum calibers but again, are more expensive than what you seem to want to spend.
On a final note, there are more than a few 'short' 30 and 33 caliber cartridges out there such as the 300 and 338 Ruger Compact Magnums (RCM), the Winchester Short Magnums and finally the Remington Short Action Ultra Magnums (SAUM).
Best bet for a 33 caliber, readily available, and lesser expensive comparably, would be the old stand by 338 Wincester Magnum.
