short stories - Circa 50's story about a man that fights off serpent-like mutant creatures in a cave as he pursues the woman he loves and prepares to kill his rival

In the opening chapter, the man is planning to kill his rival and claim the woman for himself, regardless of her feelings on the matter. He has a fall, and then he goes back in time and awakens as himself in another body. He pursues the same woman, who is in a different body, and fights his rival. During the fight and ensuing chase, the girl runs into a cave, as does his old rival. He pursues them and is attacked by weird hissing snake like creatures who live in the cave.