Sometimes is hard get full details on embroidery. The sublimation process with embroidery maybe is alternative to try give full detail on small embroidery area like hats.
Here you can see some pictures and tests for hats.
I just create a fill tatami base in white and client did a sublimation process.
Now you can see side by side. The full embroidery and the embroidery + sublimation.
Need to say thank you for Christopher Kelly for give me a change to test it!