Big firms represent the corporations, not the CEOs. There's a huge difference.
A UChi said, family law tends to be smaller and more specialized. An alum I met at a networking event told me that he got his job with his firm through a guy he played basketball with. He's now a partner, and his firm is one of the largest family law firms in the city.
He said if I was interested in family law, I should find out when the fam law section of teh local bar assoc. met (the one here has monthly breakfast meetings), and attend. Start meeting people, asking them how they got into the business, and network. It's probably the best way to find out for sure if family law is for you, and also will help you get a job.