I have just received an offer from Bain for a consultant role (experienced hire), and a week before failed the final round with McKinsey - got very positive feedback overall, however messed up math in the final round (silly mistake under pressure), and was told I could re-apply in 6 months going straight to interviews and do some math practice in the meantime.
If I had a choice, I would prefer McKinsey due to the strength of their public sector practice, which Bain lacks. Is it worth reaching out to McKinsey recruiter to ask if there is an opportunity to arrange an additional interview within couple of weeks?