Mel, as you suggested, I twisted the Jazz bass' pickup wires, and reseated the jack on the plate. Everything is clean and solid, but the hum is still there. I've found some cancellation by tweaking the pickup volume balance, and the other bass still hums less (now the AC plug is fixed), so I guess it's "normal" single-coil hum. Like you said, it's good to have gone through everything and know it's working, pots and connections are clean, etc.
Since the amp has a bit of hum that doesn't increase with volume, it could probably benefit from new caps, but since it's only mild, I'm hesitant to mess with it (or spend money on it), before selling it.