I don't really carry a first aid kit. There are few accidents I could get into that a bandaid would fix. Either a little local pressure and the bleeding stops, or I'm in deep enough that a band aid won't fix it.
I generally have enough tools and supplies to keep my bike on the road depending on how long of a ride it is. I have considered carrying a down coat that would be able to be used like a mini sleeping bag. Perhaps also a space blanket sleeping bag.
A clean rag or cotton T-shirt might suffice if I needed to cover a really big wound.