Help me re-create Roku + Sony STRD1090 + HD598 sound for my computer

I've asked this on head-fi too - so hopefully not too redundant.

tl;dr - I love the sound I get out of my Sony Receiver streaming from my Roku Box on HD598 headphones. How can I re-create this with my computer (I already have an FiiO e07k)?

longer ...

I purchased a set of Sansui speakers with an old Sony receiver (Sony STR-D1090?) at a garage sale for $50. A few years later I got a pair of HD598 headphones (in love with them). Based on everything I read - I needed a built in Amp + Dac so I got that for my laptop. It was nice but not much (if any) better than my Galaxy S4.

One day I went home and plugged it into my Sony receiver out of the blue with my original Roku box streaming some Jazz from Pandora and I couldn't put the headphones down. I have no idea WHAT made the sound difference but it was huge. I tried hard to find sound differences between Tidal, Spotify & Pandora off of my E07K & heard nothing (probably I'm not sensitive enough to tell the difference) but I'd really really like to figure out how to recreate the sound off the receiver.

It felt like things came thru that normally didn't and everything had a fuller sound without being louder. On a good violin - when you pluck the string it reverberates for a long time - it builds to it's peak then falls off it's peak (but still continues to give echos of the sound). On a cheap one the sound is a muffled dud a bit. It produces the sound but doesn't echo truly and falls off quickly. This is really the only way I can describe the sound differences.

I have up to $500 to spend (would like to keep it a lot less than that :P). Based on what I'm reading (and have heard) my 598's are capable of producing the sound I want. I just need to get a different DAC / Amp? Does it have something to do with the 3.5 vs 6.3MM headphone jack?

Audioengine D3 or Apogee GROOVE Portable USB DAC? Maybe a tube amp (a Russian guy was recommending this to me today ...)