Let Kexp+ be the class of all structures A such that the automorphism group of A has at most cnd orbits in its componentwise action on the set of n-tuples with pairwise distinct entries, for some constants c,d{c,d} with d<1. We show that Kexp+ is precisely the class of finite covers of first-order reducts of unary structures, and also that Kexp+ is precisely the class of first-order reducts of finite covers of unary structures. It follows that the class of first-order reducts of finite covers of unary structures is closed under taking model companions and model-complete cores, which is an important property when studying the constraint satisfaction problem for structures from Kexp+. We also show that Thomas' conjecture holds for Kexp+: All structures in Kexp+ have finitely many first-order reducts up to first-order interdefinability.