As someone who has participated in quite a few interviews for one of those big companies, it's doubtful that you're being pigeonholed because of your experience. However, one thing that I do see fairly frequently is candidates whose resumes look strong but, upon interviewing, show pretty strong evidence of "big fish, small pond" syndrome. That is, they performed well enough amidst their peers, but their peer group wasn't challenging them -- and they stagnated as a result, thinking that they knew more than they did. This often comes out during interviews when candidates speak confidently about solutions that are clearly sub-optimal without much apparent awareness of weakness, alternatives, and tradeoffs.

This has been a tough one in the past-interview discussion on many of the interview panels that I've been on. Given a bigger pond, would the candidate jump at the new opportunities and learn? Whether or not we decide to go that route often depends on a number of factors, both from the candidate and the current team dynamics (what's our internal junior/senior team ratio and what's our current mentoring needs/bandwidth?)

I have no way of knowing if this describes you, of course, but might at least be worth ensuring that during the interview you're not giving the above impression.