  Let $K_{exp+}$ be the class of all structures $A$ such that the automorphism
group of $A$ has at most $c n^{d n}$ orbits in its componentwise action on the
set of $n$-tuples with pairwise distinct entries, for some constants $c,d$ with
$d < 1$. We show that $K_{exp+}$ is precisely the class of finite covers of
first-order reducts of unary structures, and also that $K_{exp+}$ 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 $K_{exp+}$. We also show that Thomas' conjecture
holds for $K_{exp+}$: all structures in $K_{exp+}$ have finitely many
first-order reducts up to first-order interdefinability.
