In truth, depending on the cap this can be quite difficult, but there are a few tips that may help make it easier to stitch round patches on caps. Frequently, the problems arise from the crown of the cap shifting and warping during the stitching process. Often, there is waving and warping as the stiff crown is pushed down to the needle plate. This shifting causes registration errors and makes it hard to maintain shapes.
Make sure your cap is as firmly hooped and as flat as possible to the needle plate. Assuming you are using a wide-format frame, use a strip of stabilizer that stretches from post to post and is clipped in with the cap. This creates a cylinder of stabilizer inside the hat to give it vertical stability. Next, make sure the teeth on the cap frame’s front strap are aligned so they are in the seam between the bill and crown in the front of the cap, and that the teeth on the side of the cap frame are resting just above the bottom crown seam. Smooth the crown so the panels are flat and the cap is tightly hooped in the back of the frame.
The second way to address stability is in digitizing. Before you run the placement line for your patch, create a ‘tree’ underlay starting in the bottom center and run ‘rays’ of straight stitching out to the inner edges of what will be the round placement line, at multiple points around the circle. Offset from each other for even coverage, returning to that center bottom each time. This spreads out the fabric of the crown, smoothing it to the outer edge and permanently attaches it to the stabilizer. After that, adhere the patch temporarily, and then, instead of stitching all the way around the patch, consider stitching up to the top center from the bottom, running back to the bottom center, then running up the other side so both areas meet at the top. While I’d usually stitch around the patch, this can sometimes help stabilize extremely unstable cap crowns.
In all honesty, some companies that make these caps with regularity and/or who attach custom shaped patches elect to purchase a post-bed manual sewing machine and employ someone to manually stitch patches to cap crowns. With so many different cap styles and construction, each has idiosyncrasies in how they run. The manual method, albeit more labor intensive, allows for more control.