I've used foam inside kick drums for many, many years. I buy "pieces" of foam from upholstery shops and then cut them to size (this type of foam can also be used to line drum cases, etc). You can get the pieces left over from various jobs for next to nothing.
I normally use a "flat" piece of foam that lays across the bottom of the drum and contacts the bottom 1" - 2" of the two heads. Once I get the sound I need, I use a small amount of masking tape to lock the foam in (no duct tape - too much residue). I'm sure you could make a "ring" but I have never felt the extra effort was worth the time.
I think foam works better than a pillow or a blanket - since pillows or blankets often add too much mass, which not only dampens the ring, but muffles volume (dampen is good, muffle is bad).