There are a lot of great basses at your price point. You must be happy with your choice, and I am sure that a lot of us will offer our own ideas and spins, based upon what we like. Please start by telling us what you are already playing on, and what you may think that you might want, based upon your initial experiences (P, J, Ray, active/passive, etc.).
Meanwhile, I'll give you a list of a few exceptional instruments to try checking out:
Squier - They are doing amazing things. You may find that a Vintage Modified or Classic series bass is something for you. With $600 to spend, you might get strap locks, a bridge upgrade, and a nice case, and still have money left over for an effect pedal.
G&L Tribute series - A JB2 or SB2 may take most of your $600 to acquire, but you would become an initiate into a great fraternity of bassists who know where the best Fenders come from.
Schecter Raiden - Very impressive for a "starter," yet versatile enough to be a step-up, or even a main gigger. 24 frets, and a great sound rooted in the J tradition, but the coil splitter capability can give you $$ sounds as well.
Warwick Rockbass - again, depending on the model, you may chew up your $600, and possibly still need more, but these things are quite lethal and versatile.
Ibanez - There are so many models in your price range, that I cannot even hope to discuss them here.
Good luck.