In my experience, CD Transports can make a big difference in the sound. Having said that, less expensive transports use cheaper mechanisms, which tend to sound the same. You’ll need to spend more to get one with a really good mechanism. Also, you want a dedicated CD mechanism, not a universal one.
I have a CEC TL-5, which uses their in-house belt drive system, rather than the standard direct drive. It has an analog-like ease and musical flow, that I really enjoy. I believe it is currently the least expensive way to get a CD Transport with a seriously good mechanism. The current US price is around $2800.
Another option is one that uses the long discontinued Phillips CD Pro mechanism, but that will cost you quite a bit more.