It is fairly easy. But you cut 3/8" strips of leather.
Then start from the inside of your project with a long 3/8 strip of leather & go thru a slot, then up & over to next slot.
Then just keep repeating that order.
Idk if a narrow feed dog & plate could help eliminate some of your stitching issues.
As mentioned the diamond point or LR point needles, don't seem to pull the stitches down as deep as a S point needle.
Why can't you add the heighth on the top of the hat pattern like you want. Then just modify the top part of the pattern/cap. It will be a bit smaller at the top, but keep the headband size unchanged.