Didn't hear anyone mention vibration control. Not sure what your CD player is sitting on, but vibrations affect CD players (along with turntables) the most out of anything and I bet (regardless of your surface and flooring) you would benefit greatly from a Townshend platform or podium combo or Symposium's rollerblock Jrs, or some other brand (I only have experience with those companies). There are low level earthquakes going on all the time that you can't feel, but your equipment can.
I was amazed what it did for my turntable so much that I took care of my phono stage and amp and speakers (speakers were also a very highly use case with platforms and rollerblocks where they fit best).