We investigate the Cauchy problem on an infinite interval for the fractional evolution equation with Hilfer fractional derivative, which is a generalization of both Riemann–Liouville and Caputo fractional derivatives. Our methods are based on the generalized Ascoli–Arzelà theorem, Schauder’s fixed point theorem, the Wright function and Kuratowski’s measure of noncompactness. We obtain sufficient conditions of the existence for global mild solutions and attractive solutions when the semigroup associated with an almost sectorial operator is compact as well as noncompact. Two examples are provided to illustrate the results.