I agree with you that Shep is very sincere and concerned for these people. So he has pressed shirts. He probably brought many shirts with him. Altho he's been wearing the same one for two days now. And he's doubling up so the sweat won't show thru the shirt as much.
my point wasn't so much what Shep was wearing, but that he had options the others on the bridge didn't .. so he wasn't being "arrogant... demanding..." etc for himself but for those who were seriously suffering. I can't fault that.