Not really, I don't think H-300 need an amp unless you're trying to flavour the sound. Probably be better of with Fiio X3 or DX50 though your iPhone 5 probably sounds pretty good anyway. You may get away with checking out the EQ apps and tweak the sound that way.
Unless you want to spend $500 on Tralucent DacAmp One, then H-300 sound amazing, not even I'm that keen.
There is one amp were looking at currently called NX1 by Topping for about $40, maybe you want to jump onboard and be a guinea pig tester with us.