What to look for in a Backup Bass?

An identical bass - I like to go from one to the other without having to think about it for a second.

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I'm a little unclear on the OP. A backup is something really similar that runs a very close 2nd place to your main player. The OP then says he wants something that offers a different tone and maybe feel.
IF it's the latter, I would suggest something like the passive P/J as it fills another large tonal zone like the original bass.