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I've more or less assumed that I would be staying in my state (Connecticut) for the majority of my working life. I've only gone to schools in CT (even law school, but not Yale) and have made several professional contacts here. My question is what would it be like it I were to move to another state a few years after I graduate? Would a few years of solid experience be enough to get work in a different state? Or would I be essentially shit out of luck?

If your question is whether you'd be able to find a job, I'd think it's certainly possible, especially if you have a good reason to move (e.g., spouse moving there) and look for jobs commensurate with your experience. The difficulty is that the longer you work in one place, the more extensive your professional network there becomes. If you move, you have to start building a new network; this is especially problematic if you need a book of business, since you'll be starting from scratch in the new location.

Specifically, I might find myself moving from Connecticut to either Texas, the Carolinas, or Virginia. I've heard those states are fairly cliquey and I would just be concerned that I would be shit out of luck with a job.

It's hard to say, because hiring there can be very ad hoc. It would certainly help if you have a compelling reason to be there (spouse) or you have some connection to that region (family, ancestral, etc.).