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. :)