I would argue that Atticus shows integrity throughout the book.
He does go against his own values, for a few minutes, in Chapter 13. This is the chapter where Atticus' sister, Alexandra, comes to live with him and the children. Though we are not told this directly, apparently Atticus has asked her to come and stay because he knows that with the trial of Tom Robinson coming up this summer, there is likely to be trouble, and he wants another adult around the house for those times when he is away.
However, Aunt Alexandra has very different values from Atticus. She is a snob. She feels proud of being a Finch and looks down on other families, and she wants to pass these values on to the children. Also, she does not approve of the way Atticus allows Scout, who is a tomboy, to behave. Aunt Alexandra wants Scout to behave like a little lady.
Atticus, on the other hand, is primarily concerned that the children learn to read well, think for themselves, and treat all people fairly.
Hence, Aunt Alexandra's arrival ushers in an era of dueling parenting methods.
There is funny scene near the end of the chapter in which Atticus tries to make himself teach his sister's values to the children.
In his lawyer's voice, without a shade of inflection, he said: "Your aunt has asked me to try and impress upon you ... that you are not from run-of-the-mill people, that you are the product of several generations' gentle breeding--" Atticus paused, watching me locate an elusive redbug on my leg.
After a few minutes of this, Atticus' change of character so frightens the children that Scout begins to cry. Atticus comforts her, and then when pressed about whether they really have to adopt Aunt Alexandra's values, he gives up: "I don't want you to try to remember it. Forget it."