Hunting handguns vary from rimfires for small game to large-bore hand cannons capable of taking the largest game on Earth. You may choose a traditional revolver or autoloader or a specialty single-shot or bolt action. One thing all handguns have in common that limits practical field accuracy is the lack of a buttstock to help stabilize the firearm. Modern hunting handguns provide excellent inherent accuracy, and the short sight radius can be overcome with optics, but the absence of a buttstock is one handicap the handgun hunter must compensate for with skill and careful planning before and during the hunt. When game is spotted, it's essential that the shooter be able to stabilize the handgun enough to release a well-aimed shot.
Rather than cover a lot of specific shooting positions, let's consider the basics of how to stabilize a handgun under a variety of conditions. Adaptability is the key to success in the field as the conditions the hunter must adapt to vary, but the basic grip and position of the body in relation to the handgun should remain the same. Forget elaborate positions such as those often used by handgun silhouette shooters, and stick to a basic two-hand isosceles position as is used for offhand shooting. The goal is to support hold points that provide stability.
I should note that the following information was not gleaned from a book or any other established source. You might say it is the result of reverse engineering. After handgun hunting for four decades, I took a careful look at what worked and why. My conclusions are as follows.
Offhand, unsupported shots should seldom be attempted at anything other than very short range. As with rifle hunting, the ethical handgun hunter will use a rest and support the firearm and key portions of his body as best he can in order to minimize the risk of failure and wounded game. There are three main anchor points that aid accurate shot placement. One is the firearm and/or parts of the hands holding it. The second is the forearms; third is the upper torso. If you can securely support all three anchor points, you are likely sitting at a benchrest. However, it is often easy to support two anchor points in the field, and two is enough for most shooting. Even one major anchor point properly supported will extend your effective range.
In addition to the three main anchor points there are others worth considering. I call these secondary anchor points. If you can support the upper arms it helps a little. If you can support your rear end it helps a lot. As with a rifle, sitting positions are usually more stable than standing positions, and prone positions are often the most stable. I cannot stress enough the importance of adaptability. I will cover the basics and discuss some methods for practice, but there is no substitute for diligent and intelligent practice and developing your own style based on what works best for you.