Hey Rupe--shooting coyotes out to long distance with SP's is truly my 1st love. The best diameter bullet i have used to those ranges is the 6.5, with the .243 being the second. I would look at building the gun around the A-Max series of bullets mostly as IMO those are the best compromise between external and terminal ballistics you're gonna get if maximizing range especially with SP's. I would stick to the straight-up 243 or 260 Rem. There are slightly improved configurations of both called the SLR [super long range] series that gives you a longer neck and less case stretch to work with [i hear], but that's more time and money for little gain. I had a SS 260 Rem. built for a rancher friend and he just loves his. He's actually using the Hornady SST bullets , and he's killing everything on his ranch with it...Get a McGowen 8T in either caliber and get it going. They are the cheapest bang for the buck barrels going...IMO.
For optics i would go straight to the 3-12x Burris LER with both bells and whistles [tgt. elevation turret and Ballistic Plex reticle]. You'll need a Picatinny rail of some sort that will allow you to move the optic as far forward as possible as the eye relief is too long for prone shooting from a center grip using a bipod. Here is how i set up my system to get the most versatility for an SP hunting coyotes like i do [walking hunter]. This is a Savage Super Striker 22-250/8T McGowen set up for coyotes--
If you decide to go for a rifle optic, i have been thinking long and hard lately on the best optic for this endeavor and i would stick to a Leupold 4.5-14x VX-III with a tgt. elevation turret and Varmint Hunter's reticle, since this system when maximized gives the best practical optic from point blank to +/-500 yds.
Now all this being said the 22-250 would not be a bad choice. I especially like the 65 Sierra Gameking bullet, and 75 or 80 A-Max, and that will net you 500 yds. on the nice days. But when the wind starts blowing the higher BC A-Max 243 and up bullets will net you coyotes you may not be able to get otherwise.