I just did the same thing you did.
I happened to see a guy on Craigslist selling an unopened set of Fender Noiseless J-bass pickups for $60.
Fender Accessories N3Noiseless JBass Noiseless Jazz Bass Pickup 2-piece Set
So I found another guy selling a Chinese Squier J-bass, with hard case, for $100. I had a spare strap.
The bass was so poorly set up it didn't play at all! I had to shim the neck with a piece of sandpaper folded over to make two layers, loosen the truss rod nut two turns, and lower the saddles 3/4 of the way down.
The tuners had very narrow posts, and were so loose they rattled. I tightened things up with judicious use of vise-grip pliers, bending the two points of the housing where they meet the gear shaft just a little tighter. But then I decided to do it right, and ordered a set of these:
4pcs Chrome Bass Guitar Machine Heads Knobs Tuners Tuning Pegs Guitar Parts
A 11/16" drill bit on the drill press (hand drill if you have steadier hands than I do), and all is well.
Those Fender Noiseless pickups are very, very nice.
The only other change, was a set of these. They were recommended by others on this forum, and I'm glad I gave them a try:
Amazon.com: Thomastik-Infeld EB344 Bass Guitar Strings: Power Bass 4 String Magnecore Set G, D, A, E Set: Musical Instruments
As for your bass, don't fix anything that's not broken. But you asked for recommendations, and now you've heard mine.