Sub-$1250 DAC w/ built-in Headphone Amp


I currently have an entry level headphone listening stack, that I'd like to upgrade for both headphone listening, but to also include in a HiFi system stack when expanding my horizon to speakers. 

Currently have:
DAC = 'Ol DAC' from JDS Labs
Amp = Atom from JDS Labs

I am a bit overloaded with the options at this price point, but I know am looking for a smoother/richer sound from DAC as opposed to an analytical/etched sound. The following DACs come to mind - would any of you recommend one over the other? Source is primarily USB w/ Tidal MQA paired w/ HD58X Jubilee's.

- Mytek Liberty DAC II
- RME ADI-2 DAC FS
- DA-Art Aquilla II DAC
- SMSL M500 DAC
- Schiit Bifrost 2 (w/ separate headphone amp? or other multi-bit DAC?)
- Other?

Further - should I expect a substantial sound quality boost from my current set up?
uwrwilke
Ayre Codex. Bit more over your budget (isnt that the case for all of us) but unlikely to find a better dac-amp for less money. If you do, let me know...I tried.
If you're lucky enough to find one: Audio Alchemy DDP-1 + PS 5. I have this combo, it's quite amazing , and includes a preamp

or DDP 2 (Elac) , think 2nd gen
I have to follow up... I received a new/used Mojo/Poly yesterday, and the Mojo is amazing. More refined than the AQ Cobalt, IMO. I have to agree with the other comments... It is outstanding for $500, or any price, really.