The HD660S2 are fine for $399, but they aren't that special. I honestly hoped they'd have more bass as a total contrast to the 800 - they don't.
Why would you think that? They are not in the same league/price. HD820 brought more bass compared to HD800. That's the same league.
I can't comment on the whole 6xx series. I never tried any of them, only S2. And I can say that they are versatile and good/great at everything. You can play any music; they are good, and you can play them on tubes or solid state amps. With the natural-sounding DAC, S2 is excellent again.
I have a couple of other headphones I enjoy a lot and are in the same league, but I can't say they are so versatile for any of them.
Why did you expect more bass with S2? I didn't. They have enough bass and are very precise, and what's most important to me is that they are so exciting and involving.
I have two other headphones at the same level Ibasso SR3 and Monolith 1060C, which I like a lot, but if I have to keep one of these three, I would keep S2 without second thinking. I would be missing these two...but S2 would stay for sure.
These things are very much individual.
The whole story reminds me of the talks about espresso and espresso machines and grinders and coffee, the level of roasts and aromas, etc. So many theories.
There are hundreds of wrong and right ways when you listen to what everyone has to say...but in the end, the thing is very individual...because we all are so different. There isn't one or a few ways; there are many ways.
I think that you made a mistake expecting something that there wasn't any chance you could find there, like expecting to find a fox with a rabbit's nature
What other headphones did you try, or do you have so you could compare to S2?
But they are double the price of S2, aren't they?