It's quite full of redundancy for me especially when you go on to repeat it. I'm not sure I'd call it a hot spring if it were unnatural and they are usually outdoors so I would specifiy "indoor hot spring" but not "outdoor hot spring".
While I was enjoying a bath in a hot spring, I noticed a bear was looking at me.
Just to clarify : in Japan and, I presume Korea also, hot springs (natural is indeed redundant) are commonly used as the supplies for "baths" - these "baths" can be placed inside or outside of a building. Some hot springs are untouched and do not have "baths" built into them. In some contexts, therefore, there is a need to distinguish between a (natural, undisturbed) hot spring and an outside hot spring bath.