One of my favorite examples that shows this character trait of Atticus is found in chapter 10 when he is called upon to shoot a rabid dog wandering the neighborhood. In the past, he was known as One-Shot Finch, which he has never even shared with his children. Because of the danger and the angle of the shot, not even the sheriff wants to take the shot but trusts Atticus to get the job done:
Atticus shook his head vehemently: “Don’t just stand there, Heck! He won’t wait all day for you—”
“For God’s sake, Mr. Finch, look where he is! Miss and you’ll go straight into the Radley house! I can’t shoot that well and you know it!”
“I haven’t shot a gun in thirty years—”
Mr. Tate almost threw the rifle at Atticus. “I’d feel mighty comfortable if you did now,” he said.
In deferring to the sheriff, Atticus shows his respect for that position. He also is quick to point out that it's been decades since he's shot a gun at all, again being quite humble considering that he was once known for his talents. When he takes down the dog in a single shot, as the sheriff fully expected, Atticus is again humble:
“You were a little to the right, Mr. Finch,” [Mr. Tate] called.
“Always was,” answered Atticus. “If I had my ‘druthers I’d take a shotgun.”
Another example of Atticus's humility occurs after Scout dissipates the tension when the Old Sarum gang comes to Tom's jail cell with the intention of harming him--and possibly Atticus as well. When they get home, Scout swears to take revenge on Walter Cunningham's son, who is in her class. Atticus quickly dispels this idea:
"You will not touch him,” Atticus said flatly. “I don’t want either of you bearing a grudge about this thing, no matter what happens.”
Atticus doesn't want his children to feel compelled to rise to his defense for a situation far beyond their choosing or control. He doesn't want to stir up feelings of animosity within them toward their fellow townspeople or classmates, even when those same people have treated him with bitterness and disrespect. That's humble parenting.
Atticus also shows his humility in chapter 18 at the trial. Mayella has falsely accused Tom of raping her, and her story doesn't exactly line up. Atticus is a skilled lawyer, but using his talents to expose the lies and the horrific life that Mayella endures gives him no pleasure. Near the end of his cross-examination, Scout notes,
When Atticus turned away from Mayella he looked like his stomach hurt, but Mayella’s face was a mixture of terror and fury. Atticus sat down wearily and polished his glasses with his handkerchief.
Ultimately, Atticus needs to provide the best defense possible to his client, and that need is more important than his desire to not to expose Mayella's ugly truth--yet he is humble enough that this exposure causes him physical pain.