I came off the waitlist yesterday and am happy to share the steps I took, though I'm sure individual circumstances dictate what works best.
The first lesson I learned is that the managers of the waitlist (who can be reached at the email address cited in your original email letting you know you've been waitlisted) are incredibly responsive and happy to talk to you about how best you can enhance your application. If you ask, they will give you an indication of your gaps if there are any. Sometimes they just aren't sure about you, so it's about engaging rather than, for example, retaking the GMAT.
I met with them when I happened to be in London and we basically had another interview, but it was much more casual and based more on their legitimate curiosity than any structure. They took notes, which were added to my file.
I followed their instructions on sending updates and shared the following in one brief letter:
1. High grades in a math course I'm taking to supplement my relative quant weakness
2. My motivation for why LBS (both personal and professional)
3. That I had competing offers I needed to respond to (from schools in the same league)
4. That I received the promotion I'd told them I was anticipating
5. I had an additional LOR submitted on my behalf by someone at my firm who's perspective on me would be different from that of my initial recommenders. (The waitlist managers had told me to feel free to send in additional LORS, but quality over quantity is critical).
Over the past few months I've also been in touch with a number of current students and believe that doing so and letting the AdComm know you learned something new that strengthens your certainty that LBS is the right place would also be a helpful sentence on an update letter.
Ultimetly everything you send in is added to your file, which is re-reviewed by the AdComm, so I addressed my letters accordingly.
Good luck! I think the advice of everyone above is great: If you want to go, just keep engaging and don't give up.