What's your budget? and your "on-going" budget.
If you are going to "mostly shoot" coyote's and plan on spending some range time as well the 223 will be the cheapest to shoot if you are shooting factory rolled ammo. If you only "occasionally" hunt deer then buy a second rifle for that or "borrow one" for those few hunts you might need one.
If it has to be a "one gun for both", then while the 22's are "legal in Ontario", the 243 is realistically the smallest you want to go.
If you are working with a tight budget and you don't want a Mossberg then you are kinda restricted to an Axis or a 783 Rem (I would go with the 783 if those were the only two choices).
A few more bucks will put you into a Ruger American, a Winchester XPR, A Browning AB3 or a Savage model 11 - the AB3 is probably the best built rifle in that bunch.
Anything more than that and you are looking at dropping a grand on a "new, on the shelf in 2016" rifle which would include a blued Tikka T3, a Rem 700 SPS, a Weatherby Vanguard II, a synthetic Browning X-Bolt or possibly a Model 10 Savage (I own a VGII in 243, so that would be my choice - but I'm biased - the X-Bolt would be my second choice).
Over a grand then others open up to you...