We dont normally shotgun coyotes around here but I still enjoy my shotguns. They are ready for when the day comes to take them coyoyte hunting.
My experience has been, Hevishot coyote loads, 3" 1 1/2oz of T shot worked and patterned best with the Carlson Dead Coyote choke. But those shels are expensive! So the best alternative I found was the Winchester Varmint X BB with the same choke. These are in my 870 Express with a 28" Carlson barrel.
My other shotgun is a 30" 11-87 Special Purpose. Its still new to me so I havent made much time for testing. But so far the same Dead Coyote choke and Varmint-X BB loads are patterning great but hit a little low.
Unfortunately you can do some serious wallet damage testing different chokes and different shells. Ive been real curious about the Patternmaster Code Black turkey choke. But man, thats a $100 bill for that tube.