The main reason Heck Tate hesitates in shooting the rabid dog, Tim Johnson, in To Kill a Mockingbird is because there is someone who can perform the job better next to him, and Heck doesn't want to miss and either cause the dog undue harm or scare it away, thereby leaving a rabid dog roaming the town.
In the story, Heck and Atticus Finch come across the rabid dog, and Atticus counsels him to put it down with a shot. However, it is revealed that, although the sheriff clearly handles guns frequently, Atticus is a better shot. So, hesitating and distressed, the sheriff hands the gun abruptly to Atticus, who calmly puts a bullet straight between the dog's eyes, killing it instantly.
Heck Tate knows that this is a critical situation. If Tim Johnson isn't shot dead in his tracks here and now, there's a real danger that someone might run into him unexpectedly and end up being bitten. As town sheriff, it's Heck's responsibility to protect the people of Maycomb, so he can't afford to take risks with public safety.
There's only one thing for it: he'll have to entrust the shooting of the rabid dog to "Ol' One Shot" himself, Atticus Finch. If anyone's capable of taking Tim Johnson out of the picture, it's him. Although it may seem that Heck is avoiding his responsibilities, in actual fact he's doing the right thing. He knows that Atticus is the right man for the job and so doesn't hesitate to give him a chance to finish off the rabid dog, thus ensuring the safety of the public.
Tim Johnson is the rabid dog who strolls down the main street of Maycomb in one of To Kill a Mockingbird's most memorable moments (in Chapter 10). Calpurnia alerts the neighbors and Eula May, the town's telephone operator, and soon both Atticus and Sheriff Heck Tate arrive. Since both Atticus and Tate show up in a "black Ford"--Atticus drives a Chevy--we can assume that the sheriff deliberately picked up Atticus on his way there. Tate is aware of the secret that Atticus has withheld from his children--that he was the best marksman in the county as a youth--and the sheriff probably decided to have a better shot handy, if necessary. Atticus probably does not recognize Tate's motives, since he does not want to take the shot. But Tate insists, and he
... handed the rifle to Atticus. Jem and I nearly fainted.
Tate, who "can't shoot that well and you know it!", merely recognizes that Atticus, who
"... if he shot fifteen times and hit fourteen doves he'd complain about wasting ammunition..."
was the better choice to kill the dog. Had Atticus not been present, Sheriff Tate would no doubt have killed Tim Johnson himself.
The rabid dog, Tim Johnson, is in the middle of the street directly in front of the Radley house. He needs to be stopped where he is, or else he might go down a side street where there may be unsuspecting people out. The shooter must have impeccable aim, however, because if he misses, the bullet will go straight into the Radleys' house, possibly hitting someone inside. Heck Tate knows that he is not as good a shot as Atticus, so he does not want to take a chance on missing in such a critical situation. He asks Atticus, whom he knows "was the deadest shot in Maycomb County in his time", to do the job instead.
An extremely humble man who accepts violence only as a last resort, Atticus has not picked up a gun in thirty years. His children know nothing of his reputation; he has never said a word about it to them. When Tim Johnson comes staggering up the block, Atticus recognizes the danger and tries to urge Heck Tate, the sheriff, to shoot him quickly, but Heck Tate, in evaluating the situation, concludes that they will only have one chance, and that Atticus is the better man for the job. At Heck Tate's insistence, Atticus takes the rifle, and with obvious skill fires off one shot. Apparently, Atticus's ability with the rifle has not diminished over the years, because he hits Time Johnson right above the left eye; the poor dog never even knows what hits him (Chapter 10).