Abstract Let 𝒦 exp + {\mathcal{K}_{{\operatorname{exp}}{+}}} be the class of all structures 𝔄 {\mathfrak{A}} such that the automorphism group of 𝔄 {\mathfrak{A}} has at most c n d n {cn^{dn}} 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 {d<1} . We show that 𝒦 exp + {\mathcal{K}_{{\operatorname{exp}}{+}}} is precisely the class of finite covers of first-order reducts of unary structures, and also that 𝒦 exp + {\mathcal{K}_{{\operatorname{exp}}{+}}} 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 𝒦 exp + {\mathcal{K}_{{\operatorname{exp}}{+}}} . We also show that Thomas’ conjecture holds for 𝒦 exp + {\mathcal{K}_{{\operatorname{exp}}{+}}} : all structures in 𝒦 exp + {\mathcal{K}_{{\operatorname{exp}}{+}}} have finitely many first-order reducts up to first-order interdefinability.