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

Arcam CD37 is a great SACD and CD player. When they come up used they are under your price. 

Here are a few CD players that I've owned that were all very good. My system consists of Krell amplification and Thiel CS6 speakers.

Marantz SA8005 - This player does both CDs and SACDs. I bought it used for around $800. It's been well reviewed and sounds very good.

Marantz SA8004 - I owned this before the 8005. It also sounded very good and I believe this model made the Class A designation by Stereophile. I bought the 8005 because it had a few improvements over the 8004 (bigger power supply, etc.) and it was the end of the series. Both models performed flawlessly and Marantz will stock replacement transports for the foreseeable future. It might take some patience to find these models on the used market because they are very desirable.

Emotiva ERC-3 - I had this player along with the Marantz SA8005 because it played HDCDs (I have over 100). I compared the Emotiva on regular CDs with the Marantz and they were so close that I'm not sure I could distinguish the difference in a blind test. These cost only about $350 used but are very desirable and hard to find. Mine sold within a day of putting it on sale. I replaced this player with a vintage Krell KAV CD 250/2 because I collect vintage Krell gear but I miss that Emotiva player.

Denon DVD-3800BDCI - This is a Blu Ray, DVD, and CD player. It sold new for around $2k but now they go for a couple hundred bucks. This is a very well built player. I wouldn't recommend this as your only player but if you want to be able to play Blu Ray audio disks this is a good 2nd player to have.

A little more than you list (but can get used about your price when they are available)
Innuos zen mini ($1250)

Can load all your CD into the zen and stream from your phone.  Buy used CD, download and sell back for more CDs.  Can stream thru it. has USB to do other sources.  Can upgrade to linear power supply in future.  

I'm old school and prefer the sound of the ancient TDA1541 chip. 
Many manufacturers used this chip, I've got several players that utilize it, and would also say to find something with an old school Phillips transport. 
My stable currently;
Rotel RCD 855 
Magnavox 460 
ReVox B226
These can all be had for less than 1k

Wow, lots of great suggestions. One question becomes "How old is too old?" I see there's a division between transport-only and regular players.