Hunting coyotes

I have not hunted Coyotes. But I have read many post of people who have (I am moving to TN where I am told they are a problem). From what I have read a good number use .25 cal. with good success. Most use .30 and up so that if you are off your mark some the greater energy would compensate for the error. I have an Airforce Escape SS .25 which is plenty powerful for hogs and coyotes. I would like to get a .30 cal for bigger game. Most states from what I have read will let you hunt small game with a .25 but deer sized they require .30 and up. 

As far as what rifle to use. That is difficult to answer. There are so many that are capable of doing the job. What would your budget be? What style rifle (traditional/bull pup). I have talked to people that have great shooting rifles for $500. Some have had horrible luck with ones priced over $1,500. I can tell you this. With price most of the time comes quality. Get the most accurate gun you can afford with the power and options that you want. I like rifles that you can adjust the power easily. This way you can turn it up when you need it. With my Escape it is a single shot and I try to make sure that I am able to kill will with my first shot. Because I don't want the animal to suffer. But it has plenty of power for game under 75 yards that I am very comfortable with. 

Practice, Practice, Practice! I cant say this enough the learning curve for an air rifle is fairly big. You want to know how to set it up. Which ammo is best. How that ammo travels at each distance and reacts to the wind. It is very challenging but rewarding. 

Hope this helps some. Good luck!
 
1st suggestion I would make is to make sure and check with the state fish n wildlife to find out what if any the minimum caliber, fps, and fpe is for Air gun hunting.
Keep in mind also coyote can call under three categories
Pest animal
Non-game
Fur-bearing
Each one of those categories can have separate laws on what is and is not legal hunting method
Personally I would not use anything less than a .35 I haven't used a .308 air rifle so it might be okay but it and any lower caliber in my book would be classed as a headshot only.
So that places the question back to your goal, are you willing to pass up a boiler plate shot if you can't get a headshot? if not then I would suggest .35 Or a .45 as a caliber for being able to do both.
Don't let the falling brick myth of the .45 put it off the table as a selection, while true for handguns the .45 in air rifles has proven itself a good long range shooter.
Even so coyote are some skittish animals and it's a challenge to get them called in under the 100 yard Mark
 
In Illinois it is legal to take coyote almost every day out of the year and with any caliber air rifle. They are under fur bearer category but have special exemption. I emailed fish and game and double checked twice. As for rifle i am not sure yet it will be the distant future. As for price i would be staying in me mid range, that really depends on how much the "family banker" lets me spend. I would prefer a more traditional style rifle with a magazine for quick follow up shots. I am just seeing what others are using as there primary rifle to take them out. They are a big nuisance in my area on the dairy farms.
 
I have gone out coyote hunting many times with an air rifle and one thing I would suggest is to choose a round that is flat shooting as possible. The less movement you make the better. Coyotes will show up at many different distances and a slower moving heavy pellet's trajectory makes it difficult to quickly range your targets and then shoot.
I opted for the .25 pellet as opposed to the .357 because of the .25's flatter trajectory. In our area, coyotes are not big and the .25 has plenty of power to take one down.
 
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I'm in a very open range where I can literally see for miles but it seems like a coyote will just appear out of nowhere. I have taken 35 yard head shot with my .25 mrod and the coyote was dead before it hit the ground. I also shot one at 30 yards in the neck which required a follow up shot and the .25 had more then enough energy. I have taken several body shots with a .357 cricket and they didn't run far for me to take a follow up shot but I have found I had messed more with the heavier .357 pellet because I had to make sure my ranges were exact. If my range was off by a few yards, I would have a mess because of the pellet's extreme drop.
 
I use an FX Impact .30 set at MAX power using JSB 50s w/a Hawke Sidewinder IR 4-16x 50 scope (set on 10x). But I never take shots past 50 yards, and I always go for a head shot. At 50 yards, my Impact delivers 42.7 ft# of impact energy (according to Chairgun). It is not uncommon for the pellet to go completely through the yotes' skulls. Shot placement is extremely important to me. That's why I chose the uber-accurate and powerful Impact .30.
https://www.youtube.com/watch?v=79-QsiDq3H8
https://youtu.be/dK0IjcrFOPY
https://www.youtube.com/watch?v=TNONcrNmWDE
 
I've taken several dozen coyote, fox, and bobcat with airguns; for coyote I've taken them with a .25 at the small end to .50 at the big end of the spectrum. My favorite guns for all predators are the .30's (.30. .308, .357) in the 90 -180 fpe range. These offer a good balance of terminal performance, shot count, and trajectory. As a rule of thumb, with the lower power guns I stay inside of 60 yards and for the higher power will go out to 100 yards, depending on the guns, ammo, and shot placement selected.
 
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Just killed a coyote at 34 yards - been trying for some time to get in a position to take a shot -
I was going to use a .22 rimfire suppressed- but at the last minute went with the Daystate Wolverine - .30 caliber 
plenty of FPE for under 50 yards - took this guy at about 3 am - a shot between the eye and ear was the trick - for a DRT !
i would say even a .22 caliber could take one down but shot placement becomes critical -
better to use a .30 caliber and up - if the distance is greater than 100 yards I’d reach for a firearm 
Kind Regards 
CK