You need a rifle that can do it as well as the ability yourself, add quality glass to provide you with a clear picture and precise adjustments. I just purchased my long range killer for distant coyotes and to use as a 600 yard f class gun. it is in 308, and I wont even try 500-600 on a coyote if I can't put 90+% of my shots in a 6" circle at 600. I'm looking for MOA accuracy or better.
Get into high winds, rain, heavy snow, angled shots and I know at least for me... I'm not looking to wound the animal so it's a no go, we can't control everything all the time, I know misses happen, I'd just prefer to reduce that chance of a wounded dog, even if it means passing him up.
This is really a gun I bought to grow with, since I am a caller, and that is why I am out, taking shots even at 200+ doesn't appeal to me to greatly. I'd rather get them in close with the call, or come back at another time.
But, I also understand the urge to just see what you can do, a little ETHICAL test of marksmanship, and that is where the 308 comes in, hopefully I can get enough trigger time in between using it on coyotes and shooting at my local club to shoot it to its potential.