If you're going to hang around here -I would say go ahead and move up to a .243/6mm it will usually outrun most .224 inch bore guns and the recoil isn't anything to worry about. The .243" bore size is also a great whitetail round -it will knock deer down at 500yds and coyotes out as far as 600-700yds or more -plus if you get a hankering to try out some 1,000-yard shooting -no problem w/the right scope and bases.
I have the same rifle as you 22-250 bdl sporterweight -when I got ready to have it rebarreled I got lazy and went w/the plain 22-250 again b/c I had dies/brass/bullets etc .
About one hundred shots down the barrel in 4 years -just bored w/it I guess.
If I could do it over I would go w/a 6 srs -i'll have to check on that name -but it's a 243win w/a longer neck -that keeps it from burning out the throat as fast.1-8" twist -105 amax -coyotes worst nightmare.
The amax IS a hunting bullet but not for thick skinned heavy body/boned animals.
At 500yds the 105gr amax will act like a hunting bullet -once- it slows down a little.
Just don't ask it to be fur friendly -laf-.
I'd recommend a #3 25" barrel -it's a mildly heavier sporterweight barrel and will help the gun lay nicely for longer shots and will swing well for coyotes that are moving .
Good luck and welcome to LRHunting.
Correction 6 SLR not ssr -read here if'n you want.