Any of the 150, 174 and 180 gr rounds should do just fine. I would lean toward the heavier weights for bear, however. On a personal note, while I've had good luck with either Winchester or Remington ammo, I generally shoot the brown box PMP 174 gr. sp. ammo. It's usually much less expensive and just as accurate. I find it at Dunhams and at gun shows, although it has been a bit scarce of late.
Unless your .303 is one of the newer commercial models, its always wise to have it checked out by a competent gunsmith.
The .303 is a great gun and cartridge. Good luck and enjoy!
Most American 303 British ammo is loaded to about 100'/sec slower than the original specs, in deference to the older rifles out there that use this round. That being said, any of the American made cartridges will serve you well on whitetail or black bear or hog. The 303 is a great all-time, all-around field cartridge for game this size or even larger. S&B makes a nice 150 grain soft point but I favor the heavier 174-180 gn loads.
Evoc One gave good advice - have the rifle checked by a competent gunsmith before youstart working with it to assure headspace is proper and the rifle is safe to use. If the bore is nice and sharp she should group quite well and give you years of good hunting. JMTCW. Mikey.
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