Sorry but those pics are a bit too close or fuzzy for me to see a problem with them. It sounds like you'd be doing the right thing with a stretch fabric, by using a cutaway stabiliser -I guess it's quite firm in the hoop and not loose enough to be pulled in at the edges? The pink thread/fleece would be a stretch fabric also? So then would also warrant a cutaway stabiliser, I'd think. Can you possibly add another pic taken a bit further away? The other suggestion that may help might be to add a basting stitch to go around the inside of the hoop to keep things firm before the design sews out.

Another option is to use a fusible so that it holds the fabric firmly in all directions. Some stretchy fabrics can be difficult to keep from having this problem. It could also be a poorly digitized design that doesn't have any foundation stitches to keep the filled stitches from pulling.

did you pull the fabrics when you hooped it ? If you pull too much, it may be the problem. Make a test stitch-out on a non-stretching fabric with same designs, then You may see where is the problem : fabric, hooping or designs. Good luck !! Céline

Try using a stretch needle as your fabric looks like knit. Also an iron on mesh stabilizer first then hoop a tear away. I personally would not hoop this fabric. Good luck. Hugs. Nan W

I had trouble stitching on a T-shirt getting the right tension and I used tear away foam on top of fabric worked really slick. Used tearaway underneath. I agree with the iron on stabilizer and not hooping. Good luck.