I once made a patch for a local gun club that I printed on linen, then used a material called fusible link that you can apply to the back of printed piece by way of heat (a hand held iron will work for this.) Made a decent backing and looked pretty decent when finished. you can get this fusible link that will adhere with bot sides, or will adhere just one side. Use the 2 sided fusible link to adhere cloth patch directly to hat surface with heat, or use the one sided fusible link adhered to back side of cloth patch, and you can apply patch to hat buy sewing instead. Just an idea.
You can get this fusible link at a sewing store in choice of strips, or sheets.
After adhering to back of printed piece, you can sew a border around perimeter if you like.