This is my experience and what I've found to be a good solution. I would personally suggest getting an external amp. With the amp, just use a 3.5 male to 2 RCA plug to connect it from the audio out on the mobo to the amp. A PCIe sound card can work, but I personally don't like it.
A sound card is a valid option. I know Asus makes one that can handle headphones of 300 ohms and even 500 ohms. However, the inside of your pc case if filled with electrical interference. That noise is easily picked up by a dac and that can make it's way into your game audio leading to undesirable buzzing noises. Sound cards can be well designed and isolated to help prevent those issues, but I just prefer to play it safe vs buying a sound card only to learn that it wasn't shielded properly.
With the 6XX, being 200 ohms, I think a wall powered amp would be the best and for that, I recommend the JDS Atom. It's $99 and for that price bracket, it can't be beat. Some alternatives are the Schitt Magni 3 or the Monoprice Liquid Spark. There are USB powered amps available for similar prices, but I find them to be lacking in power and older songs are just too quiet for my ears(ex: Nuforce uDac 3).
I mentioned the dac which is really optional. Again, I believe that the dac is the last thing you should upgrade and your motherboard dac will probably be good enough. I would only recommend buying an external dac if you suffer from similar issues like me where electrical noise is getting picked up. If you want some recommended dacs, Topping makes really good budget Dacs. The Topping D10 comes to mind ($80). Another good option is the Khadas Tone Board ($100) but it's just a pcb so you'll have to 3d print a case or something.
Final thing, there are also Dac/amp combos. A really good one is the Topping DX3 ($220). So if you wanted to get both a Dac and an amp, but wanted it to be all crammed into one solution and not deal with extra cables, then this would be a good solution for that.