Plumbing help sought

The drain in my guest bathroom sink is very slow (not consistently but most of the time) and it could probably benefit from a cleaning. However, I can't snake it out from the top since the plug is fixed. So I tried to open it up below and have had no luck. Here's what it looks like (click for a bigger image):

The gray section of pipe is where the drain plug is (it attaches at the back, you can't see that in this picture), and the only connections that aren't permanent (glued, so far as I can tell) are at its top and bottom. Both of those are finger-tight and can be loosened, as they are in this picture. But there's essentially no give at all. I can't move it up or down even a quarter-inch, and I certainly can't move it enough to get it out of the trap below. So I have no way to get a snake in there to clean it out.

With things like this, probably it's trivially easy to detach something to make it all come apart if you know where to detach it. But if you just go detaching things, you'll break something. Can anyone give me any advice? (If you need more pictures or detail just let me know what.)