In this paper, we introduce a new modified iteration process for approximating common endpoint of a multivalued α-nonexpansive mapping and a multivalued mapping satisfying condition (E′) in uniformly convex hyperbolic spaces. As a by-product, we improve the main results of Panyanak (2018) with a new and faster algorithm for two multivalued mappings. Moreover, we give a numerical example to substantiate our results. Our work is new and holds simultaneously in uniformly convex Banach spaces as well as CAT(0) spaces.