How to get a referral?
1) Leverage your network - find current MBB employees in your network and ask them for a mock interview. Ask them how to increase your chances of getting in, and most probably, they'll help with a referral
2) Ask your MBA classmates - MBBs have on-campus recruiters in top schools, and your ex-MBB classmates can refer you to them
3) Reaching out to partners in your field of expertise - If you are an experienced hire, I suggest talking to partners in your industry. You can find them on Linkedin or at various events. Ask them for a short call, describe your background, and ask if you'll be a good fit. Chances are high they'll forward your resume further
4) Case championships and company events - several times a year, various offices conduct case competitions or group case sessions in which you can participate. If you demonstrate distinctive performance, you'll be invited directly to the interview and skip the line.
5) Participate in the events, send Thank You notes, ask for a chat. After each event, there is a Q&A session where you can talk to consultants 1 on 1 or in a group. Send a Thank You Note after the event asking for a mock interview
6) Cold reach out on Linkedin - try to find consultants with whom you have something in common - School, Previous Job, industry, etc. Ask for a coffee chat to get career advice or ask if they know someone who can give a mock interview. I would not recommend cold calling partners since they are less likely to respond.