McKinsey has been using an internal platform for a while now that allows its employees to make referral submissions. Specifically, the referral enters the CV of the aspiring consultant, highlighting how she/he met her/him and what kind of relationship she/he has with this person.
The rule is one referral for one application. What I would suggest, however, in case of multiple possible referrals is investigating with them (those who have not made the referral) whether they are willing to email the Recruiters to reinforce your application, highlighting why they think you could be a good fit with the Firm. It's a no-regret move.
In general, a referral from a more senior person is more effective. So, if the choice is between a Business Analyst and a Partner, it’s a no-brainer.
From my experience, having had the opportunity to network with consultants before sending the application is a great way to understand the Firm's culture and attitude in advance. The people you interact with could easily be your team members in the future! It’s one of the most powerful ways to understand if there is a good fit with the firm and if "sparks are flying." Make sure to point this out in your cover letter!
To conclude, former employees cannot access the above-mentioned portal (but they can always email their former Recruiter colleagues if they intend to) ;)
I hope it was helpful! Feel free to contact me to talk more about it.