I haven't enjoyed the HD 650 with anything but solid state, so take what I say with a grain of salt: unless the Little Dot is really bad, getting a new amp/DAC would not be my first recommendation. My experience is that getting a better amp will give you more of the things you already like about a headphone, but it's not going to change it into something completely different or drastically better. It's kind of like going from a good high school athlete to an All Star high school athlete. Better, sure, but still not a professional athlete; not even a college athlete. (Not to say that the HD 650 is a high school athlete.) The difference between how my HD 650 sounds out of a competent amp vs. an incompetent one is generally subtle enough that I can get away with the incompetent amp so long as I'm not paying too much attention. And the same goes for any of my headphones.
If the Little Dot is really and truly a bad amp / bad pairing for the HD 6XX, I think you'd know it. I used to have a FiiO E11 that sounded unmistakably horrible with the HD 650.
The general rule of thumb is that changing headphones makes the biggest difference, followed by amp, followed by the DAC. People throw around something like 85% headphones, 10% amp, 5% DAC. Those numbers are largely arbitrary, to the point of being nearly meaningless, but I'd even go so far as to say that it's probably more like 90% headphones, 7.5% amp, 2.5% DAC if you're talking about tubes, and maybe 95%, 2.5%, 2.5% if you're talking about solid state (since solid state is so easy to get audible transparency, there's less difference between high end and low. Once you reach the point of transparency, any difference is 100% headphones, 0% amp, 0% DAC).
Also, I'm with
@TMM, that any headphone only needs enough power to reach the transient peaks in your music. Anything more than that is completely superfluous. Though most measurements I see for the HD 650 are between 99dB @1mW and 103dB @1mW. I think the HD 600 are 97dB (I think).
