If decreasing "downtime" is really a priority, you could start job applications the morning after the NC game. You can keep the renewal deadline where it is, as long as acceptances don't happen until the next day. Then cut a day of job change, too. Could just be a two day process, start acceptances on the 5AM EST cycle, and run 5 cycles of acceptances, and done. Start recruiting the next morning, and consider this also the new coach signup period, which shouldn't need its own stand-alone day after recruiting. All this cuts 3 days out of the process.
The potential drawbacks include: making it more dangerous to miss the renewal deadline, if acceptances start immediately, and people have had a chance to apply for your job; making it more likely a team that loses a coach is stuck with sim, as the tightened schedule reduces the cycles for another coach to step in after a coaching change; a tighter schedule amplifies the consequences for missing days for whatever reason, which can be off-putting to the people who "have a life".
I'm on the fence, not sold that it's a priority, or even a very good idea to try to reduce down time. It will definitely have consequences that should be thought out.