Referrals at McKinsey, as with most firms, can be a powerful way to get your application noticed, and it's great that you have two! The process might differ slightly based on the region, but here's a general approach to your situation.
Having a referral from a partner can carry significant weight given their seniority and influence within the firm. However, the referral from the Engagement Manager (EM) is also valuable, especially if they have worked closely with you and can vouch for your skills and fit for the firm.
In terms of your specific situation, it would be best to:
Reach out to HR or the recruitment team at your target office: Since you have a unique situation, it would be good to get advice directly from the recruitment team. They will be able to guide you on the best course of action given their internal processes.
Communicate with the partner and the EM: If possible, have a conversation with both of them about your situation. They might have insights into how referrals work in their specific office and could coordinate their efforts to support your application.
Usually, a well-aligned referral with a specified role might carry more weight in the recruitment process. However, a partner's referral is also a strong endorsement. If both the partner and the EM know you well and are willing to vouch for you, it can only strengthen your application, as it demonstrates consistency in your high recommendation across different levels of the firm.
To my knowledge, the system does not "stack" referrals. However, multiple referrals can still be advantageous, as they can emphasize your fit and qualifications for the role.
