I usually carry a bandanna in my left rear pocket - a classic multiuse item. While I prefer to apply sterile dressings over a bleeding wound, I wouldn't think twice bout slapping on that bandanna if necessary. It is usually red, so blood won't ruin it -also fairly decent color for signaling, good potholder, general padding, etc.

How many uses are there for a red bandanna, anyway??
Geezer in Chief