Atticus does not hunt because he prefers helping to hurting.
Scout goes out of her way to explain that her father is different from other fathers. He is old, and he cares deeply about living things. On the subject of hunting, her impressions are that he does not hunt and it makes him boring.
He did not do the things our schoolmates' fathers did: he never went hunting, he did not play poker or fish or drink or smoke. He sat in the livingroom and read. (Ch. 10)
Scout begins this chapter by telling us that her father is “feeble” and “nearly fifty.” It provides a contrast to the actions he is about to take, when he shoots the dog. Yet Atticus does not shoot the dog because he wants to kill it. He shoots it because he wants to defend his children and his town. Atticus never injures anyone or anything except in defense of others.
Look carefully at Scout (and the author’s) narrative when she introduces the comment about hunting. The very next comment is about Atticus’s defense of Tom Robinson. In no small way, Lee is comparing Atticus’s shooting of the dog to his trial case. When he shoots the dog, he is defending the town from the viscous threat of rabies. When he defends Tom Robinson, he is defending the town from the viscous threat of racism. They are both infectious diseases. In each case, the town looks to Atticus, and Atticus alone, because it cannot defend itself. He is the only one with the skill to do it.
Atticus proves to be an amazing shot, even though he hasn’t held a gun in years. He takes down the dog, and he does almost the same thing in the trial. Although he does not win, he makes the town confront the issue of racism because he takes the case to trial. Simply by defending the man, and getting the jury to deliberate at all, he does accomplish a lot.
When his children get guns for Christmas, Atticus tells them never to shoot at a mockingbird.
"I'd rather you shot at tin cans in the back yard, but I know you'll go after birds. Shoot all the bluejays you want, if you can hit 'em, but remember it's a sin to kill a mockingbird." (10)
Tom Robinson represents the symbolic mockingbird. He is a man who did nothing but try to help. He made the town see their ugliness. Atticus defended him, and although he could not save Tom, he was able to save the town. Perfectly capable of shooting a gun and hitting anything, Atticus prefers to use his skills to do good.
Atticus Finch is a man who is "civilized in heart," and who feels his superior marksmanship is a God-given talent that gives him the advantage over most living things. Therefore, he decides to shoot only when absolutely necessary (Chapter 10).