OK, that helps. It had 4mm threads so I had several screws laying around. The think is, I have no idea how long it should be. Now that I know what the part is probably called, I did a google search and found this:
That's MUCH longer than the screw I have in there. Although, I'm not even sure that's the length I need, so I'll just look for the longest one I have, add the taper, and then screw it in slowly and hope it's long enough.