The major psychological impact of the second half of American Gods is on Shadow, the protagonist.
In the second half of the novel, Shadow realizes that everything he believed to be true has been flipped. He discovers that he has been manipulated by the gods for the entire novel. They even went so far as to orchestrate the death of his wife, Laura; his time in prison; and his very birth.
This realization has a significant psychological impact on Shadow. For the first time in the novel, he actually knows what is going on and fully realizes that he can only trust himself. He becomes more independent, more paranoid, and bolder in general, acting more the hero of the story than he has before.
The reader has a similar psychological experience, as they realize they have been "tricked" and "manipulated" alongside Shadow. Readers come to understand that they cannot trust anyone or anything in the story, perhaps including Shadow himself. This increases the suspense and excitement of the last half of the book, as readers realize they don't know what to expect as the pages turn.
Overall, the psychological impacts of the second half of American Gods revolve around trust and appearance versus reality for both the characters and the reader.