If your dog has to wear the cone of shame because of surgery or an injury, fitting it on him securely might be a problem if he doesn't usually wear a traditional dog collar. Typically you would thread a dog's own collar through the loop-like openings at the base of an Elizabethan collar and then fasten your dog's collar around his neck. There are E-collars available with Velcro that hold them snugly together and in place, so with a little modification you'll be able to secure an Elizabethan collar onto your collarless dog.
Place the Elizabethan collar around your dog's neck, overlapping the sides and pulling it tight enough to be secure, but not so tight that your dog is uncomfortable.
Draw a line with the marker on the bottom piece of the E-collar along the overlapping edge of the top piece of the collar.
Remove the Elizabethan collar from your dog and lay it out flat on a table or countertop.
Lay the Velcro on the E-collar, matching up the long edge of the Velcro with the edge of the collar. Use the scissors to cut the Velcro so that it is the same length as the edge of the collar. Peel the Velcro pieces apart.
Remove the backing from the adhesive on one piece of the Velcro and apply it to the inside edge of the side that will be on top when you overlap the edges of the Elizabethan collar. Be careful as you press the Velcro in place, as it may not stick as well if you have to pull it off and redo it.
Take the backing off of the adhesive on the remaining piece of Velcro and apply it to the outside of the Elizabethan collar, lining it up with the line you drew with the marker. Press the Velcro into place.
Put the collar back on your dog, pressing the Velcro strips together to hold the collar securely in place.