The ATH M50 has a sensitivity of 99dB / mW. Or at least I assume that's per mW instead of per volt, because it's the only reasonable assumption. It's impedance is a nominal 38 ohms. This means 195 mV for 1mW into 38 ohms (voltage = sqrt (power * impedance)).
So, sensitivity at 1 volts will be 10 * LOG ((1 volt / .195 volt)^2). (Since power goes as V^2/R, and dB goes as 10 times LOG of power ratio).
That means an increase of 14 dB, so the sensitivity of the M50's at 1 volt is 113 dB.
So far so good. Now the iPhone 5S will put .9 volts into 100 ohms. It should put something like at least .8 volts into 38 ohms, but let's be conservative and say .7 volts. That's a volume of 110 dB! WAY TOO LOUD!
I can't believe that you wouldn't be able to get your M50's loud enough out of a 5S. And in point of fact, I used to own the M50's when I used my iPhone 4, and I could definitely get the volume loud enough with plenty of room to spare. And, the 5S is just as loud as the 4. So both my back of the envelope calculation and my experience say that you will have enough volume.
Oh . . . wait . . . I just re-read your post. You're wondering about the 5C. Since that has a different audio codec, I don't actually know! Sorry! But just in case the above info might be helpful, I'll leave it.
What I can say is that you should be able to listen to "your" music (via iTunes samples) at a local store, with your M50's, and a demo 5C. If the 5C gets loud enough on the iTunes samples, it will get loud enough on whatever files you use for your library, FLAC, 320kbs MP3, whatever. Shouldn't matter.
As for sound quality, I can't help you. All I can say is that you probably want to audition both phones in person with the headphones you intend to use with music you are familiar with.