Considering getting a CD player (again). Used? New? Criteria?

I would purchase a universal player with a digital output so you can add a DAC to improve sound.  There are a lot of DVD-A and SACD disc floating around, they do sound very good.  A universal player will allow you to dip you toe in or indulge as your mood suits.

Various OPPO models are obvious choices.  There are several Marantz models that both sound good, and represent good value- specifically the special editions and Ken Ishiwata (sp?) signature editions.

I have a Pioneer Elite DV-79i for when I want to spin silver discs.

I still own one CD player, an Emotiva I picked up for when I wanted to audition new music before deciding whether I wanted to keep it (I use a network drive for my digital music rather than streaming).  Have had no issues with the Emotiva.

Most "vinyl-like" CD sound I've heard comes from Audio Note (UK). I got a Level 4 CDT plus equivalent DAC (so way above your stated bracket) ... BUT the more affordable Audio Note transports and DAC's (which are really quite reasonably priced) have the same vinyl-like "DNA" found in the Level 4/5 units and which I find so appealing. Worth an audition at least IMO.

As a few others have suggested get the Emotiva ERC-4 it fits your budget, performs way above its price point in terms of audio quality. Sounds much better than my Oppo 203 but will add IMO to get the most from this player it should be used with the balanced outputs feeding a fully differential balanced pre-amp. The ERC-4 is a fully differentially balanced design. I owned the ERC-3 previous to this and feel confident in stating the ERC-4 sounds better although it uses thee exact same DAC chip, the Analog Devices 1955.

Given my experience with the ERC-3 I thought a lot of people were making a mountain out of a molehill regarding the noisy transport. To that I should also add in all fairness the ERC-4 can be significantly quieter but on some disc most of the time but for what ever reason I can still hear some chattering/noise with the ERC-4 on a few but not all disc which is easily remedied by hitting the stop button, then play again. Alternatively one can eject the tray, then close again and this will fix any noise. 

In conclusion the audio performance of this player is equal to if not better than my Holo Spring DAC Level 1 using Aurdirvana + 3.5. They sound different but after a few weeks of comparing I tend to lean toward the ERC-4 which sounds more pleasing to me. Its very dynamic spacious and smooth. Highly recommended.

Marantz CD6006

Price / performance it is one of the best. Buy new and get warranty, sounds great on my system.