  We define the over-exceptional lattice of a minimal algebraic surface of
Kodaira dimension 0. Bounding the rank of this object, we prove that a
conjecture by Campana and Corvaja--Zannier holds for Enriques surfaces, as well
as K3 surfaces of Picard rank greater than 6 apart from a finite list of
geometric Picard lattices. Concretely, we prove that such surfaces over
finitely generated fields of characteristic 0 satisfy the weak Hilbert property
after a finite field extension of the base field. The degree of the field
extension can be uniformly bounded.
