I think changing Jitter settings is only option you have if you want to use Points. Could it be possible to use other visualization such as bar chart?
jthi_0-1673515338987.png
Instead of
jthi_1-1673515346777.png
You can use Customize menu to change Marker's Jitter settingsin more detail (especially Axis and Jitter type)
jthi_2-1673515640595.png
jthi_3-1673515715976.png
Also going through Essential Graphing from JMP Help page can give some other ideas
-Jarmo