Unfortunately no, there are no publicly accessable "free" hotsprings in Chaffee county. In fact as far as I know all the worthwhile springs in the San Luis Valley are privately owned as well. However, if you head to the San Juan's or the Elks there are a few different choices.
Personally I prefer to rent a cabin for a weekend that has private hotsprings - usually close to Mt. Princeton,
Salida Colorado Hot Springs Vacation Rental - Alpine Hot Springs Hideaway
Antero Hot Springs! Salida and Buena Vista Colorado.
or just pay the fee and go to Valley View or Joyful Journey.