I'm not sure who really got started hand gun hunting but, Elmer Keith probably was the best known. However, in 1967 when Thompson Center introduced a contender line with interchangeable barrels, hunting with a hand gun really became popular.
The biggest question I get from people wanting to start is, "Do I use iron sights or a scope?" Both have advantages and disadvantages.
Advantages in iron sights: It's much faster to get lined up on your target with open sites than finding it in a limited field of view with a scope. When hunting in heavy brush for dangerous game, it's easier to focus on, and swing with both eyes open on moving game. Any one who has hunted hogs or mountain lions know how fast they can move.
Advantages for scopes: It will collect more light on overcast days and low light situations. Sight alignment is much more precise and bullet placement is that much greater.
The effective range of either will not increase a persons distance ability that much. In my opinion a scoped handgun is best suited for hunting. Especially for an individual who's eye sight is not what it use to be.
But be sure, a smart hand gun hunter will have both.