How do you land contracts with large enterprises for web development and design?

I've been running a web development and design agency for the past year and it's been great. Revenues have been climbing and the experiences and learnings I've made during this time have been invaluable. After dealing with a series of small to medium-sized clients, and dabbling as a sub-contractor for larger agencies on limited aspects of larger-enterprise projects, I want to transition into doing web design and development solely for large enterprise companies, full-time.

Does anyone here have experience crossing over to that world? Any tips and insights you can share about how to land those bigger projects? Who are the best points of contact to reach out to within these firms? What are some dos/dont's to keep in mind?

Any and all help would be much appreciated. I like to have all my bases covered when it comes to planning these things and this is a great forum with lots of valuable talent circulating, so I'd love to hear what you think.

Thanks,

Tina