I want all followup shots to go into the same hole (ideally) or very close to it.
If your first shot did not kill it, there is little point in putting another shot into that spot. You should try a spot that is more likely to kill the animal.
Target rifles are somewhat specific to the type of competition they are used in but they generally have certain characteristics that are similar.
1. They are heavy. Have you ever tried shooting offhand with an 18 pound rifle? I have a video of my daughter trying to kill a running antelope at 100 yards with such a rifle. She never even manages to get a round in the chamber. I might post it down in the humor section.
2. For the most part target rifles are designed to be stable as the barrel heats but first round is seldom counted. In hunting first round is about all that counts. In F-class you will sometimes fire as many as twenty two shots without a break. Anybody who shoots that many shots at an animal has more than a few loose screws.
3. Competition stocks are specifically designed for the type of competition anticipated. These are seldom ideal for a walk around hunting rifle.
4. A long range competition rifle and a long range hunting rifle will usually use a different cartridge because a hunting rifle can tolerate a higher powder charge with the fewer shots taken than a competition rifle.
5. A really good long range hunting rifle will have an accuracy of less than 1 MOA. For example, for the kill zone of a deer, 1 MOA is about 1200 yards and for an elk it is approaching 2K and for an antelope that is just about 1000 yards. For competition nobody is going to win anything with a 1.0 MOA rifle nor even with a 0.5 MOA rifle. A competition rifle to win with, better be down in the 0.25 MOA range. I personally can't shoot 0.25 MOA although I have rifles that other people can shoot that well.
6. There are people who hunt with triggers less than 2 pounds but I am not one of them being as I fall down every once in a while and do not wish to blow my head off when I do (I seldom hunt with a round in the chamber anymore although I used to always have the rifle loaded). Competition rifles will generally have trigger pulls at a maximum of two pounds and often down to 2 ounces.
For a dedicated long range rifle that is always shot at long range there may be very few difference. Sometimes the Pa longrange hunters will used 60-80 pound heavy benchrest rifles for hunting but it is a specialized form of long range hunting.
Finally, a person should analyze what kind of hunting they enjoy. If you like to sit with your back to an oak tree and shoot a buck at 150 yards then you got no real need for precision. If you like to walk around and shoot what you see then that is a different rifle, but if you wish to sit on a ridge line and shoot across the canyon then that may still be a different rifle. Use the rifle that allows you to hunt the way you enjoy hunting and don't get a rifle that makes you miserable because you are forced to hunt some particular way that is not fun to you.