Best headphones to use unamplified?

I must have been lucky. My DT1350 has never had any issues. And as for placement, its always been: 1) put headphones on, 2) slide the cups back on the ears a touch (maybe 5 to 10mm?)...

 

I call that a design flaw. I have had dozens of headphones over the years. When that issue pops up it will annoy me to no end.

The introduction price of the DT1350 back then was pretty steep in the beginning. I expect better, especially when the HD25, M50 and K181DJ also exist...

 

If going for SQ only and price is not an issue, then I can suggest the Final Sonorous VI (though I got it from a MD deal for 500 bucks, now they are sold for 700). With the impedance of 8 ohm and easy to drive sensitivity of 105 db/V (I assumes that,because if it is mW it would be like IEM sensitive, which is not the case), it is the best sounds I have heard straight from a phone. Not the best in term of comfort, but manageable after swapping pads. But the sound is vastly different from the HD650 (a bit on the bright side, especially with the terrible sounding stock pads) so it's best to try first.