Southern Ontario Coyote Rifle
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You are in the Woodstock, Ontario, area. The farms in southern Ontario are smaller and closer together than out west. Shots are usually under 300 yards.
I would choose either the .223 or the 22-250. Either will take coyotes nicely, and even a wolf if you can find any. Farmers equate noise with danger. The .223 is a bit quieter. Also, if you show them the cartridge when you ask permission, they are more inclined to let you hunt their land.
Either cartridge can be equally accurate. My Stevens 200 in .223 will easily make 3 inch groups at 300 yards. I prefer the .223 because it is accurate, does the job on Coyotes (here in Manitoba we have to stretch the barrel a bit on longer than 300 yard shots), is cheaper to shoot, a bit less noise, reloads well, and recoils less. For Coyote Hunting, I load 60 grain Hornadys to buck the wind better. We shoot lots of Coyotes each winter, and this rig does it for me.
When I lived in Brantford, I used both the 222 (the .223 was not conceived at the time) and the 22-250 on Coyotes, Wolves, and that all time favorite, the Woodchuck. The 6mm and .243 are a bit big for what you want. Besides having about twice the weight in bullet, they both are noisy. When one goes off, Farmer Brown starts counting his Holsteins. One key to your pleasure is repeatedly being able to hunt someone's land. He might be reluctant to let you hunt the next time you ask.
Since you are not reloading right now, obtaining ammunition is another factor. Almost any half decent gunshop or hardware store will have 22-250, 243, and .223 ammunition on hand. 6 mm is much harder to find. .223 is usually a fair bit cheaper than the others.
Since you say "small game", I am assuming that the secondary purpose of this rifle is Woodchucks. After all, you are in one of the prime Woodchuck areas of Ontario. While Varmint type rifles with heavy barrels are used out west for Prairie Dog Towns, where hundreds of rounds are fired in a day, your type of shooting is more sedate. If you had a dozen shots a day, that would be a good day in most of your hunting situations. (except maybe in the spring/early summer when there are more young Woodchucks.) Your hunting is more stalk and shoot, and you will be carrying a rifle more than shooting it, so weight is going to be a factor. A standard weight barrel is probably better for your type of hunting. Your accuracy standards are "minute of Coyote", and not small benchrest groups. You want the first round out of a cold barrel to be on target.
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