You might want to clarify "thin neck".
Thin from front to back (depth)?
Skinny at the nut?
Narrow at the heel?
You also might want to add your preferred scale length, bridge spacing, aesthetics, etc.
FWIW, I love thin necks too and nothing can touch my Zon Sonus 5 in terms of sheer speed. And if you pick up a used one, they're in your price range. You might want to try some different neck profiles...sometimes thinner doesn't mean that the neck is faster. I have a Status Energy that has a pretty beefy neck, but it has a very flat back and it plays very fast because of it. I also have a Tobias with an asymmetrical neck that looks fat but plays fast, especially in the upper register.
If you're looking for a Fender type 4, get a Geddy Lee...hands down the best Jazz neck I've ever played. If you're going Musicman, the Sterling has the thinnest neck of the bunch. Or if you want something in between, a used Lakland 4-94.
Given your location near "Guitar Row", I'm sure you can find something.