I have a Fender Jazz bass like you and we probably are in the same ball park with respect to action. I use Rotosound 77 Jazz flat wounds on mine. They have decent tension (a big thing to the individual's touch) and can be surprisingly bright and ring through the mix, depending on how you set your guitar controls and your amp's controls. They have a wonderful feel, and when muting with your hand or fingers they can sound really woody and upright-ish (in an electric sort of way). DOn't be afraid to spemd the next year (or better, two) psyching yourself up to let it be OK to change out strings every 6 months with a new, different set. Hold on to the old ones as they won't "wear out". You can always change back if you don't like the new set. But a lot of people really like flats on a jazz bass. You might as well.