Well, at least you didn't ask for "advise" like they usually do.There seems to be a lot of confusion in your post. For instance:- You "don't have the academics" vs. "MSc at LSE"- "Want a new job" vs "about to start an MSc"- Name-dropping of BCG/Oliver Wyman vs. wanting to join a small consultancy- Strong communicator vs a few sentences in your post that need to be read about 10 times to be successfully parsed- High achiever, big picture person, top 10%, running Strategy & Development, working for a successful company vs. not sure how you fit inIt's all just a bit muddled really. I'd suggest that you:a) decide what you wantb) articulate it clearly and unambiguously (no seeming contradictions)c) decide a plan for getting itd) stick to the planAnswering the "raw" component of your question, however, I think there is a Top-Consultant guide to consultancy firms somewhere on this site which will give you a very good starting point