I use a spray water bottle instead of an iron
I carry a small (empty) spray bottle in my toiletry bag, which I use to wet down my hair with water after showering. (I'm a guy with very short/thin hair, and it dries very quickly.) Whenever I arrive at my destination, I immediately unpack my clothes and put them on hangers. Then I spray them all with a light mist of water, stretching out the wrinkles on the clothes. Then I hang them up, still moist.
By morning, my clothes are effectively wrinkle-free. (At least for my purposes; I'm not a suit-and-tie guy, but I do wear long-sleeve button-down shirts when I work.) I have never had to use an iron.
I tried the Aeronaut/packing cube combination for the first time on an 8-day trip (I just returned), and they worked great!