Even though you bought it as a backup, you've really fallen into another truth: A Fender Jazz (0r P) really can't do it all. No it's not an Ibby, it's not down-tuned for a couple songs, it's not fretless, it's not 6 strings and on and on. If you are going to have two basses on stage, it's dumb to just play one and say, "well,that other one is just in case things go wrong. People do that with two identical basses. To me my one bass would have to be REALLY "signature" for me to do that.
One of the local pros always has his Fender and his Ibanez. He always uses both and always rocks the house. For me it's often fretted and fretless or solid body and hollow body. If I don't need two basses ready to go, then the backup is in the truck or backstage (depending on my judgement as to how much of a sudden delay can be tolerated by the venue).
To me "backup" isn't about some second bass where you then say "if the worst happens I'll grab this". It's about thinking about the question: "What will I do if the worst happens and will THAT work at this gig? It's about a plan, not just some one size fit's all "insurance". Take my word for it, if your plan is too big of a pain, pretty soon you won't be doing it!
And sure strings do break and if you planned ahead enough to have spares that can be a backup. (playing on three strings is not much of a backup plan) I've had PASSIVE basses short out and quit. I've had active basses just cack it right on stage and it WASN'T the battery. (active basses with a passive mode are a big savior...too bad all my actives aren't that way) I've had the strap button screw just snap off at the bass surface. I'm sure there are more if I thought for a while.
The question is not "do you have a backup bass?" The real question is "what is your backup plan?" Decent plans may or may not include a backup bass on stage.