A decent dac amp sould noticeably improve mostly the technical performance of those hd600(which btw are said to be notorious for scaling very well when paired to a better amp) and if you notice a difference from different devices you have that indicates that you should benefit a lot from a proper amp.

Also dac is what generates the analog signal from the digital stream aka turns 1s and zeros into an analog signal and a better one should hep you avoid all the usually terrible built in circuitry so when you have the sub 10 bucks apple dongle available that should be the minimun requirement for you.

I'd say get a decent dac amp or a stack or an apple dongle with an amp connected (i got the very cheap topping dx1 (100$ dac amp that could totally drive the hd600) recently and i am very happy with my purchase- sound improved noticeably).