Networking via LinkedIn

I’m in the midst of a job hunt and have been actively developing my LinkedIn network along the way. I’ve found a few very intriguing jobs recently—and discovered I had connections in common with people at these companies. I’ve been able to establish new connections as a result. How best do I reach out to these new connections to get my foot in the door without looking self-serving or desperate? I’d actually JUST connected with someone at a company I was interested in when I got an interview with that person (coincidence in this case). But he asked me how I’d ended up connecting with him. It was honestly through a new connection I’d made—and I didn’t have a personal or professional relationship with our mutual connection (if that makes sense). I was honest, of course, and said I was intrigued by the company and wanted to build my network. In short, how can I tactfully utilize a new connection to get an “in” with their company?