Folks put way too much emphasis on "how far" and "best". An ethical hunter learns the skills to get as close as possible to game to increase the chances of making a good hit and clean kill, and doesn't take ANY shot ANY further than they are very confident that they can make on game.
That said there are far too many variables involved for a pat answer IMO as one who shot Contender's ALOT for around 40 years, and other handguns ALOT for over 50 hunting most game species in NA, shooting formal IHMSA and CAS for many years, endless range work developing loads, target shooting at all ranges for practice, plinking, etc, and much of it at long range because much of the hunting in the west is long range. Hunting confidence is earned by how well you develop your technique through constant practice in field positions, how well you've matched your load to the task, sighted in for BPR and know that handguns exact ballistics at all ranges with that load, how much experience you have hunting each species and your abilities to close the range gap on them before shooting, how well you can adapt to each set of circumstances and conditions you might encounter when hunting, and how well YOU are up to it on that day at that exact time all have a huge impact on what kind of results you can count on. I considered myself an exceptional handgun hunter that on a good day could make any shot I was confident in, but I also could make a less than stellar shot if I didn't stay focused.
Some of the shooting classes in IHMSA are excellent practice for offhand hunting, and in time shots on medium to big game are a gimme at least to 200 yards offhand with open sights if the handgun being used can kill cleanly at that range. And maybe contrary to what some folks believe, a scoped handgun shooting appropriate cartridges can be shot offhand very well to longer ranges if you have a lot of practice doing it with that firearm using that ammo. I never took offhand shots I was not confident I could make on game, left some longer ones to only with a lean and the longest to only shooting off a solid rest. Yeah, I made some amazing handgun kills at amazing ranges on big game and predators when the circumstances gave me the confidence I could get it done. But I also passed up far more and instead tried to get closer before shooting out of respect for the game. No ranges are given, they are not important and are always just accepted a brag anyway. How far is an individual thing, what each individual has earned the ability to do consistently and ethically.
I've owned handguns I could only count on to 7 yards for CCW, and a truck load of others all the way up to target specialty handguns that could get it done to a 1000 yards off a solid rest unhurried IF I was also up to it. Even so I limited my handgun hunting ranges to what I knew I could do consistently whether offhand, with a lean or off a solid rest.
FWIW-YMMV