I'm not sure if I'm even using the correct terms for what I'm doing in adding a bevel, so please correct any incorrect terms or misunderstandings with this!
Note that the bevel reaches rather far (proportionally) from the edge and along the horizontal face, but it does not reach too far down the horizontal face. I'd like to extend the bevel farther down the vertical face and less on the horizontal one. I've experimented with the various settings in the Bevel Tool and have tried the Bevel Modifier as well, and I see the settings are almost the same there.
What can I do to extend the curve of the bevel down along the horizontal face?
NOTE: In case there's a better way to reach my goal, here's what I'm doing: I'm making dies to go in a pottery extruder. Think of the extruder like a caulk gun. Instead of a tip at the end, there's a die, and when you squeeze the handle, the clay comes out through the hole in the die. (It's a lot like the Play Dough extruder toy many of us have seen.) I've worked with some different dies with extruders and it seems that the clay that goes through an extruder die with a nice bevel around the hole comes out as less brittle than clay that goes through an extruder die with a more defined edge. I want more of a bevel because I think that'll make a positive difference.