You may want to have a look at the Squier Precision Special. Has a jazz pickup in the bridge, and a precision pickup in the usual position. Best of both worlds!! The neck is quiet slim, very similar to a Jazz bass, and the bass is pretty light. Might be out of your price range, if you want an amp as well.
If you can stretch your budget, also have a look at the Yamaha RBX range and maybe the BB414 bass.