A used Marantz PM6004 (credit to Magick Man, for the suggestion!) It barely cost TOO much more than the Trends for a pretty nice 2ch integrated. No it's not the Aleph that Elysian suggested, but I suspect it'll be a great pairing at least initially. It may not be "end-game" but for the money, so long as I can figure out what to do with the massive chassis and presumably wicked heat it throws, it should be a good place to figure out what I want out of an HE-6 rig, or maybe I'll end up being satisfied enough to not feel a need to upgrade. Past experience with Marantz amps tells me I like the sound at least with speakers. They're generally a touch warm.
Only catch is it has no power-direct input to bypass the pre, so I can't feed the Lyr pre-out into it for my tubes, but I suspect going all SS with HE-6 is ideal anyway. It just strikes me as the kind of sound signature that prefers SS given my familiarity of the different effect of SS vs. tube with HE-400. I like tube with HE-400 more, but I'm betting I'd like SS with HE-6 more. But it would have been a nice option to have.
Get a used HE-6 and Trends TA-10.2SE if you want to keep it around $1k. If you're looking closer to $2k, go for an HE-6 and Aleph 3 or 30 (good condition ones go for about $900-1200) if you can control volume through a good quality preamp or DAC volume control. The Trends has built-in volume control, so DAC -> Trends -> HE-6 is a great compact desktop rig. The Aleph will give better dynamics, detail resolution, bass slam, and transparency. If you listen to a lot of classical and jazz as you mentioned earlier, I'd recommend the Aleph. Don't get the LCD-3 without demoing it first. A lot of people here think it's the bees knees, but a lot of people whose ears I generally trust were less than impressed with the LCD-3. The LCD-3 did not impress me at all.
Another option which a lot of people have said is to go straight to stats. You can get great classical and jazz without breaking the bank but I'm leery about bass-heavy electronica without confirming that you have a good headphone and amp pairing. The SR007 is a great do-it-all headphone.
HE-6 on the way! Some very persuasive HE-6 fans convinced me it was more in line with the direction I wanted to go with my summit rig. If it were my "every day listening rig" I probably would have picked LCD-3 for it's easier-on-the-ears presentation. But it's not my every day listening rig, that's already well established for me: Lyr + HD650/HE-400/D5k. This is my special treat rig, and HE-6 (or stats) sounds like a great choice for that. I went the HE-6 route. It cost a bit more than the Koss and a bit less than the Stax routes and I suspect I'll like the voicing more than either (And I know I'll like the round earcups more than the rectangles the others offer, dismal as HFM ergonomics are, at least I'm familiar with it )
I'll definitely keep the Aleph suggestions in mind. I first need to get familiar with the can on a modest piece before I can think of where I want to go for end-game with it. There's a LOT of stereo gear out there, new and old, tube and SS, monoblock and all. No need to rush into more expensive pieces. And I suspect the Marantz will be a lot nicer than the Trends, cute as it is, the power supply alone should be an audible difference.
SR007/009 I'm certain I'd love....but they come at prices I'm certain I don't