Playing in loud electric blues bands for years, I have to deal with this on a regular basis.
The biggest problem is the low end from OTHER instruments driving you into feedback. The bass' body acts like a resonator and vibrates in sympathy with the sound of the other instruments. It can also just muddy up your sound, since the unwanted resonances often clash with the string vibrations.
In some cases this works to your advantage; if you like to lock with the kick drum you may find a spot on stage where the kick starts resonating the bass. This can be really sweet
Your first defense is your location on stage. Feel free to move around to find a "sweet spot" then stay there

You can also stand so that your body is between the offending sound source and the back of the bass. Try stuffing some foam behind the tailpiece to dampen vibrations being transferred into the strings. If you are going through the PA, ask the soundman to keep the bass out of the monitors (especially the drummer's monitor which tends to get really cranked).
If stage volumes are still too high your last resort is magnetic pickups like Biesele or String Charger. You end up with a much more electric tone, but they do cut through and are much more feedback resistant than piezo pickups.