Before Red climbed the Wall, he decided to wrap the black blanket around him so that the drones going around the Outside won't be able to notice it. If he were to be seen and broadcasted in the live stream, Red was confident that Head Cipher wouldn't think twice about sending a bunch of Humanoids his way. For that sole reason, Red wore the blanket as an added precaution so that he wouldn't endanger himself further.
Climbing the Wall for the second time is a danger in and of itself after all.
