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).
You must be seen as a person of value. Why should they pay you and what benefit will they get? Figure out what their problems are start suggesting solutions (free of charge) to your contacts at the company. You want them to think: 'This person understands what is important to me and makes wise suggestions, I'd be stupid to not consult them'