My CPR instructor advised we carry a ziplock with a white cotton washcloth wrapped around a pair of nitrile gloves in the map slot in the door of the car. The ziplock keeps the oxygen from the gloves and the cloth keeps the gloves from sticking to the ziplock. You can glove up as you are getting out of the car and going for a larger kit, and the cloth can be used to control bleeding in more minor applications than the one you came across.
Great job! Thank you for all you do. 'Glad there are those out there like you!