Can you get a job at the beginning of any term or just at the start of the school year?

TLDR: Title

Hello! I know the Japanese school year starts in April, but I was curious if there are any accounts of people joining mid school year. By that I mean like at the start of a new term. Me and my wife (I know that will already make things more complicated), are just waiting till she graduates in May of 2026 and then we would be able to move to Japan. We don't plan on using a company like JET since they don't support housing for spouses or animals.

Also, another factor is the fact that she graduates early May. Would any job even really look at her application if her graduation date is after she submitted her application? Our hope was for her to graduate and then find a job that starts in September (I originally thought that was the start of their school year, not April). That is obviously the best case scenario but if that didn't happen, we would just keep looking until we found something. I just thought I would ask and see if anyone had any insight into the matter to have a better understanding for the future.

Any help is appreciated, thanks!