I'm in the same boat you are. I've got a pair of Senn HD598's and I've been toying around with the notion of getting a headphone amp/DAC but I haven't yet pulled the trigger on anything. I can tell you from experience though that my 598's have sounded much better through my receiver (Pioneer 1015-tx) than they do straight from my phone. I'm looking into the headphone amp/DAC is because my headphone port on my receiver has recently stopped functioning almost altogether
. I guess it is almost 10 years old and I beat the crap out of it in college. Regardless...
I have a Galaxy S4, not sure how the DAC in my phone measures up to the iPhone, or how you'll hook this up with the iPhone. If you are interested in going the route of the amp/DAC you'll need some manner of cable to drive the signal from your iPhone proprietary port (i think). The 3.5mm jack only puts out an analog signal that is already run through the phones internal DAC so I'm not sure how much impact that would really have on the sound quality. Not sure how you do that with the iPhone.
With the S4 I'm going to need a USB OTG cable. This cable has pins shorted in a manner that automatically causes the phone to default to digital audio output through the micro USB port. From there to the DAC and amp. I'm sure they have something like this for the iPhone.
This is all info I've gathered over the last week or so from this and other forums. I could be off on some of those points, but figured I'd reply anyway since I'm in the same situation. So far I've been looking at two headphone amps in particular that have gotten universally good reviews from what I've been reading.
Project Ember - $300 ish range
MP-301 MK3 - $350 ish range
I'm kind of partial to the MP-301 as I've been shopping around for a pair of bookshelf speakers as well. However, I'm sure there are other good amps in that (and lower) price ranges. I'm still shopping around. Haven't spent too much time researching DAC's yet. Hope this helps.