Once you've fried your brain on the beach - you know it's going to happen - you should be good and ready for a bit of a wander. St Barths is blessed with a multitude of hiking trails, many of which are easily accessible. That is, you don't need to hike 40 miles to get to the trailhead for your 4 mile hike.
One of the most popular walks is from Petite Anse to Columbier Beach, easily reached from Hotel Le Toiny. Keep your camera ready; you never know who you're going to spot along the trail.