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Thanks, Tom and Joey.

If I simply install a new starter without changing something else, this will happen again. That's guaranteed at this point. The geometry has to change.

I have a couple of different starter shims. They're widely available at all the big auto parts chains (CarQuest, AutoZone, NAPA, O'Reilly, etc) in varying thicknesses -- 1/16", 3/32", 1/8". I didn't use either of them the last time because I was convinced that installing the engine backplate would fix the gear alignment problem.

My plan is to install the new starter with the thinner of the shims. I'll hook up a temporary solenoid voltage source, engage the solenoid, and inspect the starter throw relative to the location of the flex plate ring gear. If it looks well aligned, I'll finish the installation and give it a try.