Start small and build up. Big companies don't make all their decisions as a collective. There are lots of individual managers who have the budget and the power to hire you. Once you connect and sell one of them, then ask them if they can introduce you to other people at the company. Build their career by making them the one who made the wise decision to hire you, and then impress more people at the company.

Find a niche. If you are highly specialized in one area, then even a large company may not have a choice in who they are hiring. My company has a high specialization in software for testing hearing on mobile devices. It's not a capability that is needed on a day-to-day basis, but it's also an area where we have little competition.