In the pic, you can see the cover wrapped around the frame and a spring clip holding it in place to the left. Additionally, you can see the bends in the fabric where the permanent clips were installed into the frame, holding the fabric cover in place. The third pic shows a close up of the clip, in place.
How did you snap those clips (the ones that kinda look like an omega shape) on to the upholstery material? They are much too small to snap in with fingers and pliers will ruin the material.