Tips for waitlists from a source I wouldn't name:
"If you are extended a waitlist offer to a program that you prefer compared to the other offers you have at the time, your first response should come within 24 hours.
Write the department’s Director of Graduate Admissions (DGA). Express gratitude that you’re still under consideration, offer to send any additional materials that might help your chances, say that the department would be your top choice if admitted, attach any funded offers from other programs that you’ve received, and provide your contact information (email and phone).
Write that department’s DGA every 10-14 days afterwards, to restate your continued interest, attach any new offers you’ve received, and remind that person of the earlier offers you forwarded
...
Every cycle, we have applicants accepted from waitlists at UChicago, Yale, Princeton, and other highly ranked schools. The only way for that to happen is if you stay in regular (biweekly) contact with that department’s Director of Graduate Admissions (DGA), apprising that faculty member of other funded offers you’ve received.
Without those competing offers you have no way to come off any wait-lists, and no leverage to negotiate your financial award."