When I traveled to Shikoku and visited Dogo Onsen, I had the chance to stay at Funaya Hotel in Matsuyama. It is where the Emperor and Crown Prince have stayed when they visited also. The hotel was a nice mixture of old and new architecture.
Dogo Onsen was about 10 minutes walk away and as I went out dressed in yukata to head towards the baths, I was given a small carry basket (bucket?) with a towel in it. The tag on the basket says “funaya” ふなや for Funaya Hotel. Some others in the street also had this baskets with their own hotel names on it.
You can see the Dogo Onsen photos here.