You could get something as inexpensive as the Musical Fidelity V90 which is $299 and it will make a difference. This with a Bluesound Node ($550) would give you a nice DAC and streaming transport. You would just connect your CD via a different input. The Bluesound has a built in DAC but it isn't very good.
The other option would be to with a streaming player. The Marantz ND8006 might be the perfect device as it is a DAC, streamer and CD player in one. The sound quality is very good and the feature set is great.
I am a Musical Fidelity and Bluesound dealer. I have no relationship with Marantz.