Glad to hear you had a good chat with the Partner!
No, you don't need to ask for the referral again. What I'd do if I were you is to either:
- Ask for a short chat with HR to see what sort of role would be best for you and get some clarity from them as to when they might open recruitment again for experienced hires
- Keep checking their website over the following months for opening and in case nothing comes up, check in with HR in about 3-4 months.
Since you already have a referral, you don't need one more. You just need them to be ready to hire.
In the meantime, diversify your application strategy to include more firms. Even if you may not want to be hired by some of them, this will be good practice for when you'll have the McKinsey interviews later on. You can use the guide below for that: