Forget the stabilizer. If you are quilting a block at a time, try hooping just the backing then adding the batting and the topping square. You may like to use a touch of spray or drops of washaway adhesive, or use a large stitch to sew an outline around the edge of the embroidery area to hold it all together - I do pin mine mostly if I know they will not be in the way of the needle.
If you are tackling a whole quilt, I just use A4 paper in the hoop, or sometimes a stabilizer. Then pin/glue/stitch around the edge, turn it all over and trim the inner section of the stabilizer/paper away. I sometimes use a sticky stabilizer for this, but it's only good for 3 or 4 uses and the tacky is gone. Some use that rubbery drawer liner in this way. When you un-sew the holding stitches or un-pin, you shouldn't need to re-hoop often.
Hope this helps