I would say spring time for hiking, before the Texas heat sets in.
I will say spring for hiking and camping but if you would like to swim on the river and in the falls/wells I would prefer summer.
Anytime late Fall/ Winter to late Spring for hiking. Great wildflowers in the spring. Swimming is nice in the summer. They have the white bass run in the spring as well if you like fishing. We go just about anytime of year. Just be prepared for the weather. Summers are so hot and winters can get freezing by the river.
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