The tail stock is just a 4-by-4 cut to fit between the lathe rails.
I was worried about how to get a Morse taper center into the tail stock. Luckily, I discoverd an amazing device – a 1″ outside diameter metal cylinder with a female #2MT drilled into it! Just the sort of adapter I need to make this lathe somewhat future-proof in terms of center options.
So just a 1″ hole bored into the tail stock was all it took. I found the center by tapping the tail stock body against the drive center in the head stock.
A second hole was bored into the tail stock where it meets the bottom of the lathe rails and then chiselled to accommodate the wedge. The wedge was just an offcut from another project. When tapped into place it holds the tail stock very firmly.
I picked up a live center from Amazon. I didn’t mention in the Head stock section that the drive center is a Shopsmith one that is set screwed to the 5/8 shaft. I got it from Joes bucket-o-rust and it fits and works exactly as I had hoped. The photo below shows a small piece of wood held between centers. Note the extra length of wood I glued to the tail stock to make it thick enough to house the Morse taper adaptor. A small hole is drilled from the back so I can tap out the adaptor if needs be.