What a great idea!! I have used my dog's lead for strays in the neighborhood, but never thought to keep a small one with me when away.
I bet your vet has these real small nylon ones, like the one that I keep in my BLeecker. Mine uses them to take a dog from exam room to lab. They have the vet clinic stamped on. You can also find them really cheap on ebay. I keep a bigger, sturdier one in my vehicle....