I'm not so sure that Alexandra ever changed much during the story, but she does show a motherly side in the final chapters that was missing throughout the rest of the novel. After Jem and Scout are attacked by Bob Ewell, Alexandra blames herself for not realizing that the premonition she had earlier had involved the children. She tenderly helps Scout out of her crushed ham costume and repeatedly calls her "darling." Then, Alexandra did the unimaginable:
She brought me something to put on, and had I thought about it then, I would have never let her forget it: in her distraction, Aunty brought me my overalls. "Put these on, darling," she said, handing me the garments she most despised.
Alexandra also seems to have changed a bit by the end of the Missionary Circle meeting (Chapter 24). She finally recognizes Calpurnia's value to the household--the guests love Cal's food--and she is grateful to Maudie for standing up for Atticus after Mrs. Merriweather insults him. Were these the turning points for Alexandra? Did she turn over a new leaf? Possibly, but prior to these two episodes, little change had been seen in the woman obsessed with heritage and "Gentle Breeding."
Her attitude briefly changed when at the end of the book the children were attacked and she gave Scout her overalls instead of her dress.
Atticus was hopeful for Tom's trial and his appeal because, even though he knew the people in Maycomb wouldn't do the right thing, he thought that the Cout would and appeal the trial because it was very clear the Bob Ewell raped Mayella, not Tom.