The usual complain with embroidering on knit caps is that despite of the thickest font and increased pull compensation - it stills sews thin.

There are several ways to skin this cat - you can try to increase the density of the stitches by about 25% and use double zig-zag underlay - which is fairly easy in Wilcom Embroidery Digitizing Software.

You can also create a base or background with light tatami fill and then place the logo on it - this light tatami fill has to be sewed with same color thread as the cap to make it subtle.

You can also try using Solvy - it is something many people use on things such as fleece etc. it works extremely well on knit caps and keeps embroidery from sinking into the fabric and maintain the width of the columns.

You can also use a combination of Solvy and stick on backing - try to slightly stretch the fabric on the top with Solvy till you are happy with the layout.