Back in November of 2001, I decided to fix the non-moving axis on the seat mechanism and see what I could do about the noise.  It was a pretty simple operation, although there were many little steps.  Here is the whole thing photographed so that you can rebuild yours with confidence.
'66 Seat Mechanism Rebuild
Here is the rust on the carpet.
Plenty of soaking with WD-40 was required for disassembly.
Note how the retainers hold the cable tubes to the frame.
Two phillips head screws hold the drive cables to the frame.  Note the square drive.
You can see the cable ends, the retainers, and the tubing.
Here is the main mechanism.
I labeled the screw drives.  The front and back are the same.
Once the frame is removed, all that's left is the seat. Here's where you would recover it, etc.
A small half-inch bolt in each corner holds the frame to the seat.
You can see how the parts go together.
See the green corrosion on the tubing? That's copper plating.
I tagged each tube to keep track of them for reassembly.
The colors are for reattaching the wires.
Everything has to be tracked carefully.
This shows how the harness was taped to the #5 tube.
I documented each tape location for authenticity.
With the tubes removed, you have to keep track of which one went where.
Inside the transmission: solid blue peanut butter (grease)
Luckily, there wasn't much rust.
Here is the problem: A sheared nylon gear.  Probably due to overdriving at the end of travel.
Exploded view: it was pretty easy to reassemble
Using a drift to drive out the roll-pin.
Here is the jack screw design.
Painted bracket, repainted motor, and a cleaned up transmission.
Sandblasted tubes: Copper plate protects the steel tubing better than paint alone.
Cleaned up jack screw parts.
Reassembled carefully on the floor with an unrestored guide up top.
The final assembly pic.
You can see how the screws are used to adjust position.
I took some liberty in the colors.  I think it I got the jack screw colors wrong.