The Key to Referrals

First, a former colleague from your previous company has joined a large corporation.
Second, your skills are widely recognized.

You started at a very small company that was unable to hire enough people, so they didn't have strict degree requirements. You performed exceptionally well and quickly rose into the top 5% of the industry. Through work, you met more people, and some of them who joined big companies could potentially refer you, helping you transition into a large corporation through the social recruiting channel.

This is one pathway.