I own a .303 enfield myself. I have never used it for hunting but I have a friend who also owns and hunts one. He has never had any problems that I know of. Several years ago he shot a big 3x3 blacktail at 50 yards one shot dead. Federal makes a couple good loads for the .303. I would stick with a 180 grain Nosler. Before hunting with the gun check and see if it is a number 5 action. If it is, it is better to "unsporterize" the gun. Number 5 enfield actions are rarer then 3's and 4's. As long as the gun has not been drilled, you should be able to keep the value. If you hunt open sights be warned the .303 peep sight has a tendancy to drift after several shots. My best 50 yard open sight group of ten shots is smaller than an lemon. That was with the federal 180's. You can drop any deer out to about 150 yards with no problem. Elk is all about shot placement. It will kill an elk if the shot is good. Sportco carries the ammo. Most larger retailers do also. Both federal and hornady make a "magnum" or high energy load too. Just make sure your action is up to snuff before using the magnum ammo. High chamber pressures mean bad news if the action is weakened. You can also order surplus military rounds online at Sportsman's Guide and Cheaper than Dirt. Just beware of corrisive ammo. You will need to clean the gun completely after shooting that. Have fun with that gun. The Enfield action is one of the best around.
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