The great thing about handgun hunting is variety, I've hunted at close range with a 4" S&W, and at 190 yds with an Encore, and I've hunted about every gane I was hunting with long guns.
If you're hunting medium ranges for medium sized game like deer, a 44mag 6" revolver with a good optic is hard to beat and reasonably priced. One of mine is a 6" S&W power-port I use a 2X Leoupold and a Burris red dot on it. I load 240 XTP over H110 for it, the gun groups 5" at 100 yds, but I limit it to about 75 yds. (Lets face it most shots in WNY are 50 yds) This makes a good all around gun if you're getting started and can't afford to have three or four guns yet...
Also for medium ranges I killed a lot of deer with a 10" TC Contender in 44 mag 240 gr to 300 gr bullets, but really 240 is enough for Whitetails.
If you're into long range over a green field for deer I've used both Contenders and Encores. My Favorite is the a 15" encore in 7x08, I load 120 Barnes X bullets, and 140 Nosler ballistic tips for it over Varget. Recoil is manageable, and it groups about 2.5" at 200 yards off bags. I've never had a deer walk more than 10' when hit with either bullet.
(Recoil is a concern I see too many people buy heavier recoiling calibers like 30-06 in this gun and the it beats them into flinching, the 7x08 won't)
Close range is my favorite, my ground is thick swamp. I carry a 4" mountain revolver & 240 gr Gold dots. But most of my close range deer with a revolver have been taken with 5.5" Ruger Blackhawk & 240 gr XTP over H110.
You need to upgrade most guns to better sights, you have to see the front sight in low light.
I've also hunted Deer with other calibers .45 Super, 460 Rowland, 10mm, but as we drop down in power levels with these calibers we need to reduce the distance we'll shoot and limit the angles of the shots we take.
The beauty of handgun hunting, I carry the encore in a backpack I built with bags, crossed sticks, range finder, seat, food, etc. And a revolver in a cross-draw holster. I'm prepped for about any reasonable range I'll need to hunt.
"The other game"
I hunt small game with a handgun, I often hunt with a 22 version of what I hunt deer with, .22 10" TC Contender, 410 TC Contender, Ruger Single six, S&W 617, Kimber .22, .223 contender, etc. It's great fun extends the season, and keeps me shooting.
