I wonder if the jacket didn't have a light-colored lining and he just reversed it. Possibly even a reversible jacket.
I'm more interested in the witness. If she thought he looked like he'd been in a fight, what gave her that impression? One thing? A combination of things? Muddy. Did she see blood? What else would indicate a fight? To me, holding a cloth to one's face. In the case of a gash or broken nose. If it was something like that, he was holding his face as if injured, he was actually trying to COVER his face? Another indication of injury/fight, perhaps he was walking with a limp. Does he have an unusual gait? Often theorized from the released clip from the bridge. We only have a snippet of her interview/description. It could be quite telling, what she saw. Blood/mud might be hard to distinguish from a distance so maybe, combined with how he carried himself, lended to her impression.
JMO