You don't have to hoop the quilt block. You can hoop sticky stabilizer in the hoop. Lay the quilt block on top matching center points. I attach in corners with straight pins making sure it will NOT hit anyway on design. Embroider design.
I do the same with tear-away stabilizer as I don't' like using sticky stabilizer. Sometimes I stitch a large basting block around the embroidery design that comes out after stitching design. The basting block holds everything to the stabilizer.