If I'm in an overstrength MOS and want to reclass to a star MOS, why do I still have to wait for my reenlistment window? Can I reclass early?

While most of these replies may have been correct at one time, policies for reenlistment and reclass change every year.
You can submit a voluntary reclass to your Career Counselor. However, there are many limiting factors. You must have at least one year Time on Station. You have to qualify with your line scores, clearance, rank etc. You have to choose an MOS at a lower strength than your current one. If you're OCONUS your report date must be 60 days following your DEROS. You'll have to reenlist to meet the SRR for your new job, so your RCP is a factor.
A voluntary reclass can take 30 days, so that one single class seat you found may be gone by the time your request is processed. Class seats are the biggest limiting factor.
Your best bet is to apply to an application MOS. They are harder to get into, so there are more available class seats.

Look in AR 614-200, para 3-19.

If you are still under your original contract, you must wait until your re-enlistment window. If you are past your initial 3 or 4 year obligation, you can re-class anytime you want. All you need to do is find a unit that is willing to accept you and send you to school for the new MOS. Typically, there is a requirement that you have a year or more on your current contract after you finish school.

In any event, you should contact your S1 or, preferably, a Career Counselor to get the most current and accurate information.