There's not a single answer to this question and it depends a lot.
If you look at where top MBA candidates work pre-MBA, you will see that only about 10%-15% of them come from MBB, so you don't need to do strategy consulting to be admitted at a top program. It would help, but I guess if you don't have top grades it is a long shot.
I would recommend you to go through official stats (LinkedIn also works) to see where MBA candidates come from. But in a nutshell, a good chunk of them come from blue-chip companies (e.g. P&G, Unilever, Nestle, Shell, IBM, Intel, etc) and have had a relatively good performance there (in many cases, with some form of global exposure and managerial responsibilities).
Assuming you don't have an unusual background that can help you to stand out (being a top athlete at university, unusual involvement in philanthropy or politics, or even coming from a rare developing country -yes, this helps!-, etc), I would recommend that you try to find a good role in a well-known blue-chip company. While Deloitte, Accenture or AT Kearney are great places to work for, I honestly think they won't necessarily help you get into a top business school. Schools want a diverse student body and they can get most consultants they "need" from MBB (having said this, there are many cases of top MBA candidates coming from "2nd tier" consulting firms).
Of course, as I said before, it all depends a lot and there are a lot of exceptions. Just FYI, and to give some "validity" to what I am saying, I am an MBB consultant and an INSEAD graduate who was also admitted to LBS (and so I know well the background of the candidates from both schools). Most of my colleagues come from top MBA programs and I have just asked around to get their opinion (which is very similar to mine).
I am happy to help and answer any further questions. Again, this is my opinion and I know too well that it is only that, there are many ways to get into a top business school (but believe me 80%-90% of the students come from MBB or blue-chip companies).