https://screenrant.com/cobra-kai-hawk-miyagi-do-eagle-fang-no-reason/

Though Hawk was given a clear reason to switch from Cobra Kai to Eagle Fang, his reason to leave Eagle Fang for Miyagi-Do is a bit more subtle.

I liked that they had him join Miyagi-Do, as it allowed him to become more balanced in fighting and his demeanor. I also think that makes him the only character who officially joined all 3 of the main dojos on the show.