Say for coons? coyotes? --- rattlesnakes and copperheads ----
I won't mention bobcats and bear -----
There is no single firearm that can do it all. THere are always compromises, as with any tool.
However, for up close ranges, it is hard to beat the 12 ga. shotgun. With a variety of shells and shot weights, you can do almost anything with it.
For the largest creatures, magnum slug and sabot rounds will put some fearsome firepower into your hands.
For the medium critters (including human aggressors), shot in the range from 0 buck through BB shot will take them out cleanly and humanely.
Smaller critters are handled with game loads in the midrange, shot sizes from BB down through 7 or 8.
For longer range, then a rifle is in order. Something in the 6mm class will do for nearly all chicken killing critters except bears. This includes popular calibers like the .243 WIN. Another great choice is the Marlin lever action in .357 Magnum, out to 125 yards.
For larger animals like bear at almost any range, you need more whollop. Heavy calibers and heavy bullets are in order.
I'd consider 30-06 the minimum rifle caliber for bear, and then only with 220 grain bullets. Some people favor the big bore lever actions in calibers like .444 Marlin good bear medicine. I'd agree.
Pistol suggestions possibly too. (I also trailride)--Mainly pistols seem to be to big in the grip for me comfortably. I have wide square hands, but short fingers.
I'd recommend a 4"- 6," double action revolver in .357 magnum as the best all around handgun. They are not overly heavy, but are one of the heaviest hitting handgun rounds going against human targets. For anything else but bears, it's a lead pipe cinch.
If bears might be encountered on the trail, then the best thing to do is leave. Shooting bears with a handgun from horseback is not well considered.
Nonetheless, if you want something heavier, then a .44 Magnum is a decent choice. There are more powerful rounds, but they are not common and there is a price in weight and recoil. Beyond the .44 Magnum, practicality becomes a consideration, as well.
Both of these the ever practical .357 Magnum and the powerful .44 Magnum will take some training, as they are LOUD with sharp, brisk recoil.
However, both can be stoked with their little brother rounds... .38 Special in the .357 Magnum and .44 Special in the .44 Magnum. This gives reduced noise and recoil and them excellent choices for general trail and pot use. I'd have to again give the nod to the .357 Mag for this use, as the .44 Magnum/.44 Special combination is a lot heavier.
Although in good leather, either one can be carried comfortably. as for grip size problems, the solution is after market grips. I also have smallish hands, so I use finger groove, strapless Pachmayr grips on my K frame .357 Magnum. They - and similar grips from other makers - can be had for almost any revolver made.
It should be noted that both the .357 and .44 Magnums are also chambered in repeating rifles, which adds to their versatility. Having a rifle and handgun in the same caliber has long been a welcome combination when one is "out-a-ways."
Self loading, semi-automatic pistols are not the best choice, generally, unless you may encounter human assailants in numbers. For animals, and general use, their speed and ammo capacity are not really needed.
They can be cranky about ammo selection and fiddly to operate, although quality, modern auto's are fairly free of these issues. The worry over jams and stoppages exists, too, but that can be minimized with ammo choices. However, it can still happen and it is more than an inconvenience if it occurs at the wrong time...! There are none of these problems with revolvers.
Auto's are not normally as accurate as revolvers, either. And, except in certain cases, they are not more powerful than equivalent, common revolver rounds.
On the mean streets of L.A., they may be just the ticket. And they do look good in the movies and thug rap videos.
But, on the back trails and around the farm, holster a DA revolver and get good with it.