Usually Bain&Co. hires via their On Campus drives and through their off campus drives as well. If they come to your campus well and good. If not, then be active on Linkedin as their employees regularly post about different job openings. You could also ask for referrals from your seniors/ any personal connections to expedite your application. Most importantly keep practising cases and guesstimates for such a role.
For getting in a Consulting Firm through off-campus recruitment drives-
Consulting Firms do hire via on-campus placement drives is true but they only hire via on-campus placement drives is not. Mostly all of the consulting firms do also have off-campus recruitment drives/lateral hiring going on throughout the year.
For the same, keep an eye on the LinkedIn job portal, their career pages and LinkedIn posts by the employees at these firms. One thing that could help here is tapping people for referrals!
Since consulting is more of a 'Generalist's Job', experience in any and every industry would be relevant and in fact, diverse industry expertise is always better for the firm!
Besides, it is never too late. :)
We, at Seekho, are here to help you on your journey.