SAN FRANCISCO—Uber Technologies Inc., in its search for a No. 2 to Chief Executive Travis Kalanick, is interviewing candidates with track records in large, established companies—a sign the ride-sharing titan is looking to temper Mr. Kalanick’s idiosyncrasies in exchange for a corporate culture more typically in tune with its size and ambitions.

Mr. Kalanick built a defiantly competitive startup culture at Uber over which he reigned supreme, turning it into a global concern valued at $68 billion but also leading to a series...