I see what you mean. We don't know the exact moment Taki decided to change schools but I think she made that decision before she met Crychic.
I think the timeline went something like this:
- Taki joined the wind ensemble club in middle school.
- She was hit with the hard truth that she couldn't escape her sister's shadow.
- We don't know how long she lasted but at some point she quit club, gave up on the trumpet and picked up the drums.
I think the moment she decided to quit and find something else to do is the moment she decided to change schools as well. It's her attempt to start a new chapter in her life without Maki's influence.
- Taki then joined Crychic in her third year in middle school.
I think it's a cruel form of irony that even with Crychic, she was recruited thanks to Maki's influence. She was visibly annoyed when Sakiko and Soyo started talking about Maki, which is understandable... As I said, Taki probably already made her decision by this point. If that's the case then her introduction to Crychic only helped affirming that decision.
- Crychic disband and Taki changes to Hanajo with the start of the new school year.
It's possible that Taki came to the realization that Crychic (including Taki herself) was reliant on Sakiko, and that gave Taki the drive to become more independent. She started learning to compose, and started working at Ring to learn more about the inner working of the band world. So it could be that this is the time when she decided to start a new chapter in her life, and that included going to a new school where less people know about her and her sister.

There's no final answer for now, but I'm more inclined to believe she made up her mind before crychic. We'll see.
There are several things related to Taki that we simply don't know yet.
When did she quit the wind ensemble club? Was her sister supportive of her decision? Did her family approve of her changing schools? How did she convince them?.. It's a mystery for now.
Hopefully we'll get some answers next season. Even if not to everything related to Taki, at least some of it.
(And please reveal Maki's design so I can draw her properly.)