Lathe Spider

[Toolmaker51]

Nice job, indeed. I particularly like the wedge design. I'm going to add that one to my 'appropriated ideas' file.

And, of course, you realize that you've now made yourself a chuck depth stop. A simple rod will suffice but you can titivate it by drilling and threading the chuck end of the rod so that smaller diameter inserts can be used when the work diameter is less than the rod diameter. (If I were doing it I'd make the rod about 3/8" diameter.)

A 3/8" rod is a great start, will accommodate many material shapes. Depending on headstock length and RPM range, most diameters will go harmonic and whip inside the spindle tube. The best cure is a nylon or aluminum bushing to maintain central position.
My preferred stock for a stop rod or ejector is light weight tubing, with bushing. Accepts 'tooling' just as easily,to compensate different forms of turned work pieces.

I rarely edit my posts.......(more than once a day). Warranted in this occasion about stop rod tooling.

Whatever material selected, find a lightweight long compression spring of diameter suitable for a nice sliding (non-binding) fit. With any kind of setscrew collar as a seat, it will eject parts from chuck or collet by varying the compression, a little nudge or completely out.
It also adds tactile sense when contacting the rod, upon loading the next part.