Now for the coyote hunting I do praise my 25-06 with a 100grain bullet, there just ain't no better in my way of thinking. If you are speaking of the Bobcat well my 222 or 22mag has served me well. Those cats are usually a lot closer anyway when I put a bead on them and drop the hammer.
As far as hunting cats I can tell you what not to use. I took my 30-06 with me on my lion hunt, I never got to fire it but one problem with the rifle on a cat hunt is all the climbing up steep steep snowy terrain. It is hard enough to keep yourself from falling let alone keeping a rifle on your back. My rifle has some good battle scars on it from my cat hunt, as I fell a few times. Next year when I go I am going to get some sort of hand gun that I can put in my pack or on a shoulder holster. CUBAN
For lions I prefer to carry a pistol....22-mag or bigger. If your calling in coyotes...I really like a 22-250 or 223...they are super fast and do a great job without really tearin up the hide....course if you don't care about the hide...any rifle is good..lol
I carry a .44 mag for bear and lion and a 22-250 for coyote. I like carrying the pistol as opposed to a rifle for the same reasons Cuban gave. I carry the 22-250 in the truck only, if I see a yote, I whack him, or try to anyway.
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I like my .220 swift for coyotes. It does an excellent job w/ no hide damage. I like to use my bow on bears. I haven't chased a pack of hounds though, only hunted them from tree stands. I am sure I would like some kind of pistol for chaseing hounds.