Floorstander reco given a few criteria...

OP, it seems like you’ve opened up a lot of possibilities by saying $4k over floorstanders and an amp. (I think that’s the gist.).

I like jjss49’s suggestions (I’ve heard Spendor; Wharfedale also sounds like a good fit). You might also consider Quicksilver, Totem, Focal:

Separate Preamp and amp ($2300)
https://www.usaudiomart.com/details/649637401-quicksilver-line-stage-tube-preamplifier/
https://www.usaudiomart.com/details/649656945-quicksilver-mid-mono-tube-amps/

Floorstander: ($1415)
https://tmraudio.com/speakers/floorstanding-speakers/totem-sky-tower-floorstanding-speakers-mahogony...
or these ($2100)
https://www.usaudiomart.com/details/649600872-totem-hawk-speakers/

Integrated amp: new, this integrated is still a great deal: ($2k)
http://quicksilveraudio.com/products/integrated-amplifier/

You could pair it with these, used and be a bit closer to $5k:
https://tmraudio.com/speakers/floorstanding-speakers/focal-aria-926-floorstanding-speakers-prime-wal...

Or these and hit closer to (4k, total):
https://tmraudio.com/speakers/floorstanding-speakers/vandersteen-3a-signature-floorstanding-speakers... — you’d want to double check that the QS integrated would drive these adequately. Vandy folks here would know.

Thinking a bit more now....I bet the Quicksilver + Spendors + your phono setup would sound really lovely. You could tube roll a bit to find the sound that suits your taste, music, and room. Search for Quicksilver in these forums -- they bat pretty close to 1000 with folks.

This is excellent, thanks JJ, AudioTroy and Hilde!!! I have quite a bit to research, awesome. 

OP, just a P.S.
The Music Room is used and gives 14 day trial. They verify that all products work. They’re great to deal with. Trying in your own space is crucial.
Crutchfield has free shipping and very inexpensive returns (e.g.$10 for 160lbs of floorstanders). They carry some very good brands, including Revel, Focal, Dynaudio, Wharfedale, Klipsch, KEF, Martin Logan, and more). Their open box deals can be amazing. I used to associate them with only having mid-fi or lower end stuff, but that is not true. Amps: also have some good ones: Anthem, Arcam, Cambridge, McIntosh, to name a few.)

What system sound and performance characteristics do you value most, and what would you most like to improve upon over what you have now?

I like detail and separation. Right now the sound is kind of hollow and compressed in a way. All the notes are there but they aren't crisp and like distinct or spaced out, if that makes sense. If there was a goal it would be warmth and distinction or space.