You seem like a very strong candidate!
If I recall correctly, Deerfield, and most others, don’t open their application or their interview calendar until later in the summer. You can call and ask about the timing if it is important to you. When they open their calendars, they only open a few weeks at a time. So, if you are looking to interview in November, you might not see any November openings until September or October. It’s a real pain for those of us who don’t live within driving distance of the schools and need to plan flights and hotels.
One recommendation I got from many AOs: don’t hesitate to call a school’s admissions office and talk with the receptionist or whoever if available. They aren’t likely to make notes about your call in their files. If they did, they would probably be impressed with your interest anyway. If some unknown thing is stressing you out and you can’t find an answer on cc, don’t be afraid to call.
I also agree with skyline29 that you should consider applying to other schools. While watching my kids go through this process over the last few years, I was surprised how much luck is involved. A lot rides on the interview, and if you connect with the interviewer, that can make all the difference. On the other hand, if you are an artsy kid and are interviewing with the football coach, it might not click. Much of the application is about building up your application, but there is also an element of luck. You can mitigate that risk by applying to other schools.