Bandaids
Running with the theme of not wanting to publicly bleed on myself, I make sure that, no matter what, my travel kit contains a few bandaids. More often than not, I end up giving one to someone who just got a papercut. Also, if you’re in a pinch then can serve to temporarily hold the bathroom stall door closed if you happen to get the type that has especially strong adhesive: when you find yourself in a truck stop and don’t exactly want to meet the locals in a vulnerable position in the bathroom, one tends to get creative.