Cool! I like u/carnaxcce idea if you don't want to fool with waste canvas and stitch on Aida, then trim and adhere to the felt of your choice!
Fray Check can be helpful on the edges of the Aida if you go this route.
Don’t quote me on this because o haven’t personally used it (I’m a believer in leave enough around the piece you can comfortably cut off the frayed part later) but I’ve heard this stuff works well. I give you the Amazon link, but if you’re US based, Michael’s and Joann’s usually stock it.
Absent rolling that hem up and encasing it there's not much you can do. Depending on how visible the area is use a some thread and just do a couple stitches to hold is closed and then use some fray check to keep it from getting worse.
https://www.amazon.com/Dritz-674-Liquid-Sealant-0-75-Ounce/dp/B0019KBWA2
Put some fray check on the edges of the hole, and sew on a patch. The patch has to be of material that is similarly stretchy to the compression gear, or its not going to stay (so those iron on patches from the arts and crafts store, sadly, will not work). The easiest way to make such a patch is to cut up other worn out or ill-fitting compression gear.
There are a couple of things I've used for this, often in conjunction. If you go to a store that with a sewing section you can find a product called Fray Check. This stuff works very much like the glue method that Mungo mentioned, it's just specifically designed for that purpose.
The other thing I often use is Fusible Fabric Backing. This shit works great for thin or flimsy canvas/fabric style patches. Basically you cut a sheet roughly to the size of your patch and then iron it on to the back of the patch (you'll most likely need to trim the edges again once you do). This not only locks down the edges, but gives reinforcement to the entire patch. Having said that, it will make the patch a lot stiffer than it was originally (assuming you put it on a canvas/fabric patch). So if you're trying to maintain a look of thinner/flimsy/flexible patches, you'd want to avoid this method.
These links are only for example purposes. You can find Fray Check a lot cheaper in a store, and unless you're manufacturing jackets in bulk, there's no way you'd need that much backing. I bought two yards worth quite a while back, and I still have some to spare.
It looks like youre just embroidering on stabilizer. I used to make tons of patches.
I used a semi thick NON STRETCH cotton duck cloth type material, and 1 sheet of stabilizer. You can then use anti fray stitch glue (not a ref. link, straight google) https://www.amazon.com/Dritz-674-Liquid-Sealant-0-75-Fluid/dp/B0019KBWA2 and put that on the border. Wait like 5 minutes. Cut out around the edge of the patch. Try that :)
I use dritz fray check on serged seams. To make it really durable though, I suggest going back to the corners where you backstitched and stitch again with a straight stitch, then use the fray check.