We study the existence of subshift covers for topological dynamical systems, the infimum of the entropy jumps to such covers, and various aspects of conditional entropy and covering maps including a variational principle for covering maps. In particular we show every asymptotically h-expansive system (and therefore by Buzzi every C ∞ homeomorphism of a compact Riemannian manifold) has a subshift cover of equal entropy. Our arguments in dimension zero are extended to higher dimension with theorems of Kulesza and Thomsen.