Tube recommendations

6H23 is a Russian tube type but a 6992 can be substituted. These tubes are very revealing and have an extended top-end. 
  With a phonostage it's important to get tubes that are low-noise and tested for microphonics. The 6992 is a tube that can become microphonic.

It seems as though the 
Voskhod 6H23 is a very in-demand tube. I would roll the other tubes and keep this one for now.

https://www.head-fi.org/threads/schiit-lyr-the-tube-rolling-thread.549508/page-559

Infection
I believe shadorne may have the tube substitute number for the 6H23 incorrect. I believe the correct tube is 6922. Another good source is Upscale Audio.com. they also have good user reviews on tubes and good selection. 
I have never had problem with any tubes bought from them.With regards to Telefunken tubes vs Gold Lions. I have used both companies 12ax7 tubes in my phono stage. The Gold Li on tube is best value for money but they are not as good as the Telefunkens.

I  am a tube roller myself .  I just got an old Antique Sound Labs 845 SET integrated with multiple 6SN7s and a 6L6GC or KT-66.  I rolled all of the except the power tubes (845s) but I am seriously looking at replacing them.  I know my tubes pretty well and figured out which combination of 6SN7s would yield the sound I wanted. 
I agree that you should listen to your phono stage to learn what your baseline is.  You never know you might be happy the way it is.  You should really know what sound you are seeking as well/  Unfortunately in your situation you need to find tubes that are dead silent. Truly silent tubes that are fairly old are not that easy to find. Phono stages pick up and magnify the smallest of signals and do the same with any noise that is present.
The other matter is knowing where you can stretch things .  For instance my favorite 12AX7 or ECC83 is neither.  It's a 5751, however 5751s have lower gain than a ECC83.  If you are only using MM cartridges you might get away with that roll.  If you need all the gain you can get then you can't. If you want to keep the sound sound clean and clear then Telefunken is a good choice, some would advocate for current production.  If you want to warm up the sound then try to find a Blackburn Mullard from as early a period you can find. There are several other levels in between.
  One of the best 12AT7 or ECC81 was  made by Brimar it's a short lived sub brand, premium tube, they  called Trustee Brand and the ECC81 tube was numbered 6060, they are still available in the second hand market.  Andy doesn't like Brimar so don't tell him I told you that.  They are excellent tubes in MHO. 
The most difficult and expensive tube you will be chasing is the excellent E88CC or 6922.  They are expensive , if you like a clean crisp sound try to get a Siemans or Tlefunken.If you choose to go with a pro like Andy at vintage tube services be prepared to spend some money.  Another excellent tube pro is Brent Jesse at Audio tubes .com (he is on vacation) There is another great pro tube seller who will generally be less expensive - Jim McShane find him on audio asylum at McShane Design.  I am not affiliated with anyone yada yada yada,

Excellent comments, @mechans .
Very true about Andy and Brimar, he's knowledgeable and honest, but there are certain tubes he won't sell.    
I placed an order recently and went elsewhere for the Brimar I wanted. Really outstanding tubes.