Streaming virgin

@dagery Does the kindle have digital out? If it does you could go digital out to the USB B input on your integrated amp.

If not, you could purchase an Apple iPad (or any pad that has digital out ) and use the Spotify app and go digital out from the PAD to the USB input on the integrated. This will be a good way to get your feet wet without spending to much. I have an iPAD that I do this with using the Qobuz app on the iPAD. If you buy the iPAD air I think it has a USB C connector instead of lighting connector. USB C to USB B connectors are easy to acquire. My iPAD has the lighting port. I just bought an adapter that goes from lighting to USB A then has a USB B termination that will go into my DAC.
The only negative is the PAD is hard wired to the integrated and runs on battery during this time. The advantage is that if you do decide to invest in a dedicated streamer you can always use the iPAD as a controller.

The other advantage is if Spotify lets you download tracks.  Qobuz has a feature where in you can download HiRez files local do the Tablet/PAD and then play them.  In fact its the required method to play anything higher than Redbook CD quality.  Can come in handy if you have planned network outage or trouble shooting network SQ issues.

Example
AudioQuest USB C to USB Bhttps://www.audioquest.com/cables/digital-cables/usb-b-to-usb-c/usb-b-to-c-forest

That integrated has an 1/8 inch mini jack input on the front of it. Have you tried using that with the kindle? you mentioned you were using earphones. Of course this will be using the crappy DAC in the kindle.

Is there a DAC within the integrated amp or transport?

You may already have a DAC and only need to have a streamer to your system.  

@mrklas
Yes that Amp should have a ESS9018K2M according to the user manual.  That is why I was recommending going digital in.

Then as a subsequent step you could fire up a trial subscription of  Tidal or Qobuz and try hi rez to see if its worth the spend. Deezer is also an option and streams at RedBook CD quality and has a vast music collection.

OH!  @dagery if the Moon 260 is a "D" it should have a DAC as well.  Does the CD player have digital inputs on the back?  Or was the unit $2,000 or $3,000?