My advice is to pursue further degress only if you are certain you want to specialize in that field.  

The only LLM with which I am familiar is one in tax. I am interested in tax policy and considered taking a job with the tax court and pursuing a tax LLM simultaneously.  I ended up choosing a district court clerkship instead. I felt that I got more benefits, more door openings to a greater variety of things this way. But that's reflective of my uncertainty about specializing in tax exclusively. If I were certain that was what I wanted to do - then the tax intensive year would have been much much more beneficial to me than this more general credential.

I don't know much about other LLM programs.  I imagine that an LLM from a highly ranked law school may be helpful to those whose JD is from a school with less local or national prestige.

Okay, so the title of my comment is a little harsh, but I am serious.  Here's the thing, you've been in school 7 years or so to get into a profession that you likely don't know for sure that you will even enjoy.  While it is daunting and a little scary to start your legal career, I do not think that adding another year of law school to the equation is a good idea, in most circumstances. (How about that little disclaimer!)  Get to work; see if you like being a lawyer; figure out what you really enjoy about being an attorney; break free from the protections of full time school; live!

My perspective is, as always from a biglaw firm associate, and the truth is that law firms could care less about an LLM.  Even the tax attorneys in my office do not have tax LLMs.  However, I also know that the tax department doesn't have any junior associates so if that is where you want to start out, a Tax LLM might open that door for you.  With that said, law firms do appreciate other advanced degrees that line up with what are you may be practicing in.  For example, law firms go crazy for people with science or engineering graduate and post-graduate degrees who can do patent work; they also appreciate MBAs.  I could imagine other types of practices where a specific masters degree in something like home land security, public health, education, etc, would be beneficial -- but not so much an extra year of law school.

With all that said, if you have no job prospects and getting an LLM from a better, more connected law school could open up some career doors, then I would consider it.  Be careful though, that if you get a tax LLM (for example) that you want to do something related to tax.  Employers can see through somebody that just did an extra year of law school to put off going to work or to get access to a better career services office and will be suspicious of a person that got a tax LLM but wants to work as a public defender as her life long ambition.