I'm not sure I agree. With due respect, small firms run with even tighter profit margins than big firms. Grads are not sufficiently cheap to keep on the books, and their limited skillsets mean that they're very hard to charge out.
This is true in a large firm as well as a small firm, but larger firms tend to sell larger projects -- and can squeeze a grad in to do some of the grunt work. Smaller firms generally sell smaller projects -- 1/2/3 people at most -- and it's much harder to bring in an unskilled graduate to do the work, especially at the sort of fee rates they charge.
Finally, one of the reasons that the big boys take on graduates is that they have defined training programmes to give the new consultants the skills that they need. In a smaller firm, even if they're willing to take you on, you're unlikely to get the kind of training that will turn you into a more well-rounded consultant later on.
If you're not having luck with the big firms, but your credentials are good enough to get you interviews, it suggests to me that you need to work on your interview techniques. Failing that, I would suggest getting a few years experience in industry and trying again later -- at which point both big players and smaller niche firms should be interested in your skills.