Yeah, I also never carry any type of first-aid kit. Perhaps I should. About the only thing I would have with me to use would be an ever present buff, my belt, or shred a shirt or shirt sleeve if needed.
There is a place I fish often. It is a high steep bank just down stream of a bridge that is covered with large stones to prevent erosion. Now and again I feel like I was lucky not to do a head first dive to the bottom as I scramble down closer to the water. Though most of the stones are the size of a coffee table all of them are not stable, and will sometimes take an unexpected tilt. Giving me a chance to hone my balancing skills.
Duct tape is a favorite do it all product. And when I was working I always had electrical tape to patch up the occasional cut. Working in hospitals people were always pushing me to go to the ER. What, and have them giving me a large bill for a applying a band aid. Ah, no.
The self-fusing silicone tape might be a good multipurpose type tape to carry that might work better than duct tape. AKA stretch and seal tape. [ the same stuff used to make a Sebata Yūzō type kebari , [瀬畑雄三さん風毛鉤]. It is stretchy and the self fusing feature may seal up a wound more effectively.
Getting the wind knocked out of you is a scary experience.
Not something you want to repeat. At least I did not. When I was maybe 12 y.o. I was alone and fell out of a tree, panic set in quickly when for several seconds I couldn’t get my lungs to suck in any air. I still clearly remember the experience.
Plentiful initial bleeding probably helped wash away a lot of bacteria lessening the odds of getting an infection. A good thing as long as the bleeding can soon be stopped.
Hope you are on the mend. 