I found that when this was happening I had to spray starch it several times to make it quite stiff before I did the embroidery.
You have it framed nicely. You could try ironing from the center to the edges with a stiff fusible interfacing to help straighten out some of the puckering. For most of my projects I wash, dry, and iron the fabric. Give it a good starch and then hoop it with stabilizer and float another piece underneath the hoop. Using two stabilizers on most projects gives a nice finished look.
Do you float the same type of staplizer?? And is it just placed between the feed and the hoop??? And, starch?? Liquid or spray???
Any type of starch will do. It's just about making the fabric firm and usually a tear a way stabilizer yes between the feed and the hoop.