I recommend taking a look at the Topping G5 if you can stretch your budget to $299. Slim case with good clean power output.
Note that the iPhone will require some sort of Lightning to USB adapter to connect to a portable DAC / amp.
I also agree with other posters that if you're going to try to listen to music on a plane, you really need active noise cancelling (ANC) headphones. I have a pair of AKG N400NC for travel. They sound great and ANC performance is decent. If you're interested in buying, I'd wait for Black Friday. They've gone on sale for as low as $39, I believe.