I think your plan of getting a headphone amp first then a DAC later is a great plan. In general a headphone amp will get you the first big advance.
Good choice of headphones.
I have spent about 15 years upgrading from headphones connected to my headphone system now. You can see my system by clicking on my ID. I have owned a dozen amps. While solid state added power and slam, they always lacked real musicality. Lots of details, but without natural midrange…. without bloom. Then I got my first Woo… WA6 finally natural sound! I had some very hard to drive headphones (unlike yours), so I got an WA6SE (still makes all headphones sound better… power required or not). Then finally I went for a truly spectacular system and got a WA5… holy cow! An enormous jump up on all headphones, efficient or not. Woo… only way to go for me.
In the mean time I upgraded DACs… each move made an big improvement. With power conditioning and finally my Ayre DAC I really got magic. Just barely… but it was there. Finally I got a real streamer, Aurender N100… holy cow… incredible system… I’m afraid a PC just isn’t the environment to create high end sound… a real streamer is required.
So, a truly great headphone system requires a great streamer, DAC, and head amp as well as headphones. Each well chosen component will raise the level of performance… you probably cannot imagine how high performance is possible… I sure didn’t know. The great thing is my headphone system is only 10 -15% the cost of my main system. The problem was, my headphone system got much better than my main system and I ended up spending a small fortune to upgrade my main system… I am so lucky to have such problems.