there are a lot of things you can upgrade. For the bridge baddass2 is nice, provides a lot more sustain. For the pickups, can get quarterpounders, basslines, ultrajazz etc. You can consider upgrading the preamp too. It makes a lot of difference.
I'm guessing you don't really know what you're after - what exactly do you mean by more sustain?
Also, what do you mean by "dead as night" when you turn to your bridge? Do you mean to say that your bridge pickup isn't working? If so, just check whether there's a broken connection somewhere. Pickups don't break unless you've tried to pull them out forcibly or mess with the copper winding or something like that.
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- because tone matters -
sustain is the rate of decay of the note. if the note decays quickly, the sustain is considered to be long. you are never gonna play a 16 second note in reality, but even for shorter notes the sustain is noticeable.
consider the fact that the harder your attack, the shorter the sustain. so be gentle on the right hand and you could have better tone. after all, the tone is in the fingers
Don't be caught up in a rush to upgrade or change stuff to your guitar. Pickups wise, you should have a general idea what type of sound you are going for. For my precision bass, i upgraded the pickups to SD Quarter-Pounders only to realize its a bit too 'rock' for my liking ( Got SD Antiquity for my Jazz which I LUUURVE). Installed a Badass II Bridge to my P Bass too and the sustain can go for days heh. Anyways, to summarize, take your time to research and be definite of what you want before you start upgrading. It's better than rushing to upgrade only to somehow regret it later on.
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