not sure which area you're in so i don't know if this is relevant.
But where I am, (California), private schools don't really have much the strict guidelines compared to public in regards to hiring teachers. (ie. you dn't really need a teaching credential etc). And the thing is that most of them have a grass roots way of looking for teachers (ie. someone on the faculty who knows a lot of teacher friends is in charge of getting subs).
i'd suggest try to get into subbing jobs first by asking any teaching friends if they know of vacancies or to get put on the subs list at their school. since full time teaching may be hard with full-time med school interviewing, maybe you can try to book long-term subbing jobs (this is what I did, subbed for a teacher who went on maternity leave). It's great because you don't necessarily have to write all your own lesson plans since you can base things off of the original teacher, but with long term subbing (ie. maternity leave for teachers can be 6-8 weeks, that's almost a third of the school year), you get a lot of freedom to improvise and tweak the curriculum and lesson plans. The teacher I subbed for gave me a lot of freedom, and i had a great time.
This is also good because tbh, if you only plan on teaching for one year, you don't wanna be experimenting with a teaching style and curriculum. For the students' sake as well, if you go in where there's already some structure, they're more likely going to be able to learn better as opposed to being guinea pigs.