1+ That's the approach I would use. Don't use the veneer pieces that have been stained or finished. Use only the bare wood veneers. Any bare wood veneer should be fine. You should have pretty much a lifetime supply for guitar repairs with the amount you got from your friend.
In the pic you have your spoon pushed pretty far into the slot, and it is flat rather than wedge shaped. So the first question is how far into the slot is "full length."
1) The way I would do this is by putting the spoon in wherever you need it to be to get the bridge right.
2) take a thin strip of veneer (maybe 4-5 mm). and see how much closer to the bridge you can insert it in front of the spoon. The place where the veneer is tight or an mm or two closer to the bridge is "full length." Mark the spot a piece of blue tape.
3) If you can slip the strip of veneer into the slot at the heel end of the spoon, you may need 4 rather than 3 strips of veneer to make your wedge.
4) I would probably use 4 strips rather than 3 in any case, since you want it snug and you can sand a little if it's too snug...and sanding would even out the steps a little. Just like BD suggested, but quarters rather than thirds.
5) Paint the side of the pocket and the strip with Titebond, and put the first strip up to the full length mark. Clamp it in place. Use a small piece of wood wrapped in wax paper between the clamp and the veneer so that there is even pressure on the veneer.
6) Wait 8 hours.
7) Same procedure for the next three pieces of veneer, with 8 hours between each.
8) Put the neck in and see if any adjustment (sanding or adding veneer) is necessary.
You can use the same procedure, if you want to, at the base of the neck/pickup cavity to add an mm or two of string length, though as long as you can intonate the high E to the proper length you may not need to do this.