I would suggest using the phone and give it a couple weeks then determine if you are unhappy with the sound. If you are thinking of a dedicated *portable* DAC, you might as well research and find a good lossless player with an excellent DAC or buy a DAC/AMP combo. Ugh. So many options.
My advice is to use the device to play content that suits your preference (ease of use, streaming content, radio, whatever) and use it with the E12 to drive the HE-400s. A phone and the E12 will be a tight fit into most people pockets. I can do it, barely.
I will use the phone for a couple of weeks and see how it goes. Then, I'll research the DAC/combo units you suggested and see where it leads me. Thank you for your helpful comments.