'Dressed like that' could also just mean he was wearing the DD costume when he snuck in the building, or even the black alternative. And then he's forced to change into whatever is available.

I don't know about him losing his shirt, though. He's got bruises and noticable scars and while no one there would likely ask about them outright, they're still suspicious identifying marks that Matt would be almost as reluctant to show as his face.

So. Maybe there's a robe of some sort? Possibly a little see through, so the muscles can be seen but not the scars.