Hoop hats with your regular embroidery hoops.

Step 1:

Hoop the stabilizer very tight. I cut my stabilizer the full length of the roll and leave it in one long length. It saves on stabilizer because you don't cut squares and waste most of the square. Draw your center line down the stabilizer. On the Brother hoop there are small raised notches on the hoop. Use those for drawing your line.

Step 2:
Flatten the hat peak. Just gently bend it the opposite to what it should be, until it is very flat. Be gentle...

Undo the back buckle of the cap. Turn the cap inside out, and roll up the inside flap and flip it over the brim part. It will stay there if you flip it up far enough.

Now center your front seam of the cap over the line on the stabilizer.

Pin in place, then gently push the fabric of the hat to the top, and center over the stabilizer line as you go. Don't stretch the cap, just gently move the fabric. Pin the top in place. Instead of pinning in place, you can use double sided tape to do the same thing. Just keep the tape well back from the embroidery area so that your needle doesn't pierce it. The only drawback is that to adjust the hat is almost impossible because the tape only sticks really well once. I keep my pins well back from the embroidery area, and never have to move them.

now gently push the material from the center to one side and pin..don't stretch. Gently push the fabric from the center to the other side and pin. Your hat should now be reasonably flat. Sometimes it needs adjustment, and for that, all you need to do is to remove one or mor of the pins and gently rearrange the cap material. After a few practice runs, you will be able to hoop it flat the first try.

When you embroider the cap, try to position your embroidery  as close to the peak seam as you can get without running into it. You have about a 3 1/2 inch embroidery area here, and I try to keep the center of my design no more that 1 3/4 inches above the peak. But keep in mind that it will vary with your design size.