3 month old choking on saliva/spit up in car seat placed in car seat sling in restaurant

Sarah

My baby was strapped in her car seat in a restaurant and we placed her in one of those hammocks that is meant to hold a car seat carrier. My baby started choking out of no where...face red and mouth wide open. Gagging. It was the scariest I've ever felt in my life. Right when my husband got her out and patted her she was completely fine. Nothing came out!!! So my question is was it the angle she was laying in?? Are those car seat holders not safe???