Anasthaesium
1. When you say "top", how Top is that? Trust me this question is very relevant.
2. Off cycle hiring? You are better off networking your way in than waiting for a response in that case. So if you are in a top business school, then am sure you wouldn't have a problem finding classmates employed there.
Yeah, I agree. the best way would be to actually get in touch with people who currently work their, including consultants and HR. I would also think about a story to tell why you didn't apply earlier. Did you change you industry preference? Why did you decide that consulting is a right thing for you now, given you didn't apply several weeks/months ago.