I've been a serious varmint hunter for a long time. We have 'yotes in the suburban areas of West MI now- they'll snag your pet persian or chichuaha off the deck come evenings.
First- calling and decoying- Don't over call- your set up and camo and working around shifting wind conditions is critical. No wounded rabbit or cat or other "hor's d-euver" for the coyote or fox squeals at high volume or for any measurable period of time. A decoy can work, as in hunting turkeys, but coyotes and fox are mainly scent oriented creatures.
I usually hunt with a partner- my son-in-law is "hooked" on varmint hunting now- woodchucks in the summer, fox and coyotes in winter and spring- although we shot 12 fox on private land (where it is legal with landowner OK) this past summer, evenings while shooting woodchucks as well- Right place, right time, right arsenal--
Winter we use one scoped varmint rifle (22-250 or .243 Win) and we like Hornady moly-tipped loads, and one 12 gauge shotgun- mod or full and the copper plated Turkey loads (not legal for waterfowling) are great. OO buck will ream a fox or coyote a new one, but if you want to sell the pelts, No. 4 buck or the coppered turkey loads are the "ticket"
Using a Parker 12 or 10 would be a great treat- sort of like a "blast from the past" vis a vis Wm. Harden Foster's saga of the family "little gun" the fabled 16 hammer Parker-
I shot another coyote from a tree stand opening day of our deer season, the private farm (260 acres Mecosta Co.) is adjacent to a sheep ranch- coyotes like lamb chops, but a 180 grain Core-Lokt from a older M70 Win makes "good ones out of bad ones" very quickly--