Not at FI yet, but hoping to in another 8 years. Not sure how helpful I will be. I'm a government lawyer. The agency I work for hires a miniscule amount of attorneys right out of law school, and I was lucky enough to be one of those. We do occasionally hire laterals, but usually from biglaw, and since the hiring freeze that doesn't happen anymore. However, the pay is pretty good (
http://www.opm.gov/policy-data-oversight/pay-leave/salaries-wages/2012/general-schedule/), the job security is great, and the work (I practice administrative law) is interesting. In admin law, it's a lot of appellate-type legal arguments, less factual discovery-based.
I recently tried to leave my job for biglaw so I could move to another state, and I had the same experience you had- need 3-5 years experience in a pretty specific field, and it was not mine. Since I'm an admin law lawyer, I have very little discovery experience, and was not nearly as markeatable as I thought. Fortunately my agency transferred me to said new location anyway, so I no longer had to look. But I've found it to be fairly difficult to switch fields, even after only a few years in.
One way to transition into a new field may be to take on some pro bono work. For instance, I'm interested in practicing estate law eventually, so I've been taking on some pro bono cases. That way I can learn about the field, make some contacts, and develop experience in the field. However, this is something I'm thinking about doing as a side gig once I reach FI, so I'm not really looking for the money the same way you are.