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.