I recommend you cast a very wide net. MBB's quite picky about grad institution, GPA, standardized tests, work exp, leadership. ~10% of those invited to interview will receive an offer, ~1% yield of all apps. Of all grad-level hires - MBA and APD - ~65-70% are sourced from universities with T7 business schools, ~85-95% with T15 business schools. Laying this out not to deter you but to make sure you go in eyes wide open. As such networking will be essential, assuming you have the stats they're looking for.
With that said, check out other consulting firms as Argentium listed (via Vault, job boards etc) and figure out what kind of consulting you're willing to do (market access? competitive intelligence? market research? corp strat? bus dev? R&D? tech? and so on). Each does something a little different. Bar to entry may be lower, depends on firm, but networking will always be a common thread. Some examples:
Tier 2: LEK, OW, PwC S&, Parthenon-EY, Monitor Deloitte
Big 4 + A: Deloitte, EY, KPMG, PwC, Accenture
Life Sci Specialists: ZS, QuintilesIMS, Clearview, Putnam, Clarion, Artisan, Huron, Navigant, Dedham, Xcenda, CRA, Promidian, Inventiv, DRG
Edit: just noticed you've already graduated. You don't qualify for MBB internships - need to apply for full time. Small consultancies are probably your best bet. If no luck, go back to school.