Tough request: $100 phones that sound good with jazz and classical and have good soundstage that don't need an amp. If I could name these phones, I'd own them...
You might want to try the Senn 497. They don't really need an amp, they sound pretty neutral (so probably pleasing with classical and jazz), and they have decent soundstage as I recall...Give them a listen. About $60 USD.
In the sub $100 range, you might consider going for even cheaper. My favorite sub $100 phones for classical and jazz are probably the senn mx500 earbud, although alot of people like the koss ks35. Neither of these is as clean as phones in the $65-100 range, but they are more musical than most, can be driven out of a receiver, sound reasonably neutral, and have decent sound stage.
Actually, you could probably buy all three of these for about $100.
None of my recs, though, are "semi" closed. Phones to block external noise too?...man...
In my opinion, the best sounding phones with classical are the etymotic 4s. They benefit from an amp tremendously, they cost $300, but they are the most articulate and musical phones for catching the nuances of classical and jazz music. Given the rude disregard of your price range here...I can say that the er6, which a few people have mentioned, IMO sounds a bit flat and compressed with classical music. There is a hump in the mid treble that pushes the orchestra out of balance to me. But at $125 USD, they probably sound better than your senn 570. Sorry to not be more help.