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
The Bifrost sounds like a great choice (and add a separate amp like Valhalla)! 
I have only heard a Ragnarok (an amp) but that was great so I trust Schiit as a company. They also have many great reviews. I think you will notice big improvement in SQ. I have an iFi Nano and a Dragonfly Black but prefer my separate dac through the Ragnarok. If you go with Valhalla to keep the price down you can always upgrade the amp later and keep the Bifrost.

Other brands to check out could be Okto or iFi (with their more expensive solutions). 

https://www.stereophile.com/content/okto-dac8-stereo-da-processor
https://ifi-audio.com/products/neo-idsd/
Nuprime makes some good gear too, though I think they have just introduced new models...I think some of the confusion mentioned before is that while the title states w/built in headphone amp, the OP also states he is considering a Schiit DAC with separate headphone amp...
Lots of choices here of DACs, so here’s mine. I have Benchmark DAC1 that I have had for many years. It sounds great and can be used as a preamp. The Benchmark DAC1 MSRP was $995, but can now be found for $500 or less. With your budget, you could probably find a DAC2 or a DAC3 that would meet your needs. If you are looking for a smooth sound, I recommend a tube unit. In any case, I think it makes sense to listen to as many as you can before you decide.
The Wadia Di322 is out there in the wild if your patient. Its a good DAC and has headphone capability;  2 coax, 2 optical, 1 USB inputs plus XLR. Does DSD. ESS Saber 9016S implementation. Very similar to the Mcintosh D100 I believe.

Might find one in your price range.
Caveat: Mcintosh group has shelved the Wadia line but support service is available.
The newest Cambridge Dacmagic looks pretty cool...MQA and all. I have an older one and it's great sounding. I also use a Schiit Bifrost 2 with a "Bluenode" 2i for MQA streaming although I rarely use it with headphones. Headphones in bed reading, pair of awesome Grado 325e with a Schiit Magni 3 from an old iPhone...yeah man...