Did you get your instrument fixed up?
Your description and picture are a little odd... I assume the plastic bodied jack is the one you're trying to replace, but it looks like you've got a new Switchcraft phone jack installed in the pickguard that looks like it should work just fine.
The piece you bought from StewMac will work fine too. For a mono-output instrument like yours, a stereo jack is wired in a tricky way. The tip carries the signal and the sleeve (the long cylindrical part of the plug) is the ground, just like a regular mono jack. But the other contact (the ring) is wired to the battery power. When you stick a mono plug in a stereo jack, you short together the ring and sleeve and that powers up the electronics on your bass.
The jack you got from StewMac is a little more advanced. It has a real switch (actually a pair of them) that are flipped open or closed when a plug is inserted. You use a jack like this when you really want to bring a stereo output out on the tip and ring contacts but still want to switch the power on and off. The switch connections are the other 6 conductors on your jack. If you use this jack, you can use these contacts to switch power or you can just ignore them and use the tip, ring, and sleeve connectors.
The wiring for your new jack isn't that clear from the photo. For the Switchcraft jack that you have installed in the pickguard, the audio signal goes to the top lug in the photo (it's its on the opposite side of the plug from the longest contact which has a bend furthest from the hole). Ground wires for audio and power go to the middle lug. The bottom lug will be the wire that goes back directly to the battery clip. Is this a P/J bass? That would be the best explanation for all the ground wires (looks like 2 pickups+filter grounds and that audio is separate from that, but I can't really tell).
Good luck,
David Fung
