To say that Bud's experience with the Amos family didn't go well is a massive understatement. Bud woke up at one point to find that Todd was shoving a pencil up Bud's nose. The incident turned more violent from that point forward, and ended with Todd telling a massive lie to his mother about Bud and bed wetting:
Todd said, "I just tried to waken him to make sure he'd gone to the lavatory, Mother. I was just trying to help." He aimed his finger dead at me and said, "And Look at him, Mother, this one's got 'bed wetter' written all over him."
Bud can hardly get a word in, and he ends up being locked in the shed that belongs to the family. To Bud's credit, he doesn't panic. It's a scary situation that he finds himself in, but Bud copes with the new environment fairly well. He initially has some momentary panic that builds, but he effectively collects himself and begins to take in his surroundings:
I closed my eyes and thought real hard about making my breathing slow down. Pretty soon it sounded like the five other breathers in the shed had left. I was still scared but now it was that get-real excited-and-want-to-move- around kind of scared.
One of the things that Bud sees is what he thinks is a vampire bat. This is where Bud begins to react rather than think, and coping very much involves thinking. Coping means dealing with and overcoming problems and difficulties. Those might be emotional or situational, and people have varying degrees of success when trying to cope with difficulties, and experience matters as well. People learn to cope better based on previous coping successes and failures. Bud fails to understand the importance of the "bat" not moving or hurting him. He ultimately makes the wrong decision to antagonize the "bat" which turns out to be a nest of hornets.
As Bud is thrown into more new environments, he deals with and overcomes problems and difficulties in a way that reflects his learning from previous experiences. Bud is slower to react. We see this in how he stays calm in Hooverville and when he is told that the librarian he seeks isn't there anymore. Readers see this when he wakes up in Lefty's house and fakes like he is still sleeping. Bud is figuring out how to best deal with the strange new circumstance he has found himself in, and a big part of his coping mechanism is allowing his mind to take in information rather than immediately reacting to the situation. By the time that Bud gets to Grand Rapids, his coping skills are much more refined. That's what allows him to adapt so well to the various personalities of the band members as well as deal with the fairly antagonistic nature of Calloway. Bud takes his time with the man. Bud learns about him and the band. Bud is committed to seeing his plan through. Bud's ability to cope has allowed him to stick with a situation far longer than he did back when he was with the Amos family.