I am not sure there is a lot one can do about a waitlist. if I were on an adcom, and sitting there for many years, I would have created a system to have a waitlist that is both as fair and orderly as possible, i.e. one where I would rank waitlist candidates from strongest to weakest. That way, the moment an opening appears at my school, I would always try to pull the strongest candidates first.
Where any additional letters/updates/phone calls may come into play is
1) if you have something significant that would make you as appealling as one of the stronger candidates, i.e. publications, significant award or honor, etc. But I doubt that there could be any such sudden change over the last 3 months, such projects are usually mentioned in the application earlier.
2) If the case is that you are ranked lower than 2 or 3 other candidates and you are all relatively similar in "desireability." Then if you are the only one to write a letter that has good updates and sincere desire to go the said school, then you may be chosen before the other 2 or 3.
Purely conjecture here.
