So here's my take on it, as a 660s owner myself and who has that same rig as one of my set-ups;
Try something DIFFERENT.
Something that is a different style amplifier.
What you have there is a very competent rig, but it's pretty much solid state through and through.
I like Schiit. If you want to stay Schiit, but switch up the base sound sig, you could try one of the hybrid tube amps like the Lyr.
Or you could try a true tubie like the Woo Audio offerings- I have these cans on the various Woos from the trusty and well-loved WA6 to the WA2 to the WA7 Fireflies. They all sound very nice with the 660s, and because you run a different tube set with each amp,(as well as build and specs), you get a different sound on each one. I can honestly tell you that, if you don't over-gain the 660, each amp in the range offers a nice justification. I'm not saying buy 3 or 4 different tubies off the bat, I am saying that any one of these is a choice of merit.
Or maybe try a nice solid state amp from a different builder, like JDS Labs or Grace or that Cavali hybrid is sure sweet....
I am so NOT helping, eh?
I guess where I am going with my,(somewhat sleep deprived), blather is to say that, if it's me, doing it over again with these cans, I would say YES to trying a different gear, and one that shifts the SS in a different, yet pleasing direction. There is so much out there that you can do!
I think, for a nice starter into Tube amping, the WA6 is a good way to go, literally thousands of tube combos to be tried in search of the sweet spot for you. Or a different tube set to roll for each kind of sound genre that you listen to.
If you want to try different for Solid State, JDS Labs The Element as a very nicely priced elegant and wonderful sounding DAC/amp in one object of art. It's probably my personal favourite desktop solid state amp right now.
Good luck on your search!