Question
I've heard some folks find ways to get in aggressively, while others talk about networking and "staying on the radar" until the big company choses you (because you were friendly and stayed in touch meaningfully on LinkedIn or something along those lines).
Answer
I would say the best way is to ask questions socially / track their support forums / converse with employees to discover a problem they have. If you can demonstrate that you can solve their problem and generate revenue / cut costs to a greater extent than your fees, then you stand a pretty good shot at getting in.
All companies have externally visible problems, if you're willing to look hard enough.