This is an honest question. Why would you hunt coyotes or any larger animals, inc. deer, with a shotgun where you have a choice?
Are you hunting in a densely populated area?
If you need quicker action than you think you can get with a 'scoped rifle, an iron-sighted .243 -.30-06 class seems about right. That's especially true where a wounded rabbit call may produce a cougar or a wild hog or a bear instead of the coyote you're hunting.
And a rifle reaches out better, too. A member of another board (lives in Washington state) used his .303 Lee-Enfield to whack a coyote about 80 yards from his back door one morning. A single shot hit the coyote in the head, killing it instantly. He was using old Remington RN ammo with 180 grain bullets. I guess it shoots to his sights on that No. 1 MK III. I'd prefer the Winchester or Norma ammo with spitzer (pointed) bullets, but he likes that RN Remington bullet, which also works well on deer. I think he's taken a couple of elk with it, although he owns a .338 Winchester for that sort of thing. The .338 is massive overkill for coyotes; the.303 is his grab-it-quick gun if surprise animals appear around his rural home.