You can stay at any hotel, they just have to fulfill the SHA+ requirements to enroll in the program.
By the way, SHA+ is not reserved for Phuket, I know hotels in provinces with no sandbox that are proud to show their SHA+ logo to their guests. Kind of nationwide "covid free" certification.
You'll need to come back before November 19 and you'll get another year. If you plan to leave during that second year, you should apply for a re-entry permit at the immigration before leaving.
what seems absurd to me (and to the immigration staff) is that you want to get an extension in the middle of your way. Almost everybody does it before departure or after arrival, not passing by. The only reason would be if you live in a camper, but Thais are not familiar with this kind of housing yet.