We contribute to the theory for minimal liftings of cut-generating functions. In particular, we give three operations that preserve the so-called covering property of certain structured cut-generating functions. This has the consequence of vastly expanding the set of undominated cut generating functions which can be used computationally, compared to known examples from the literature. The results of this paper are not only significant generalizations of previous results from the literature on such operations, but also use completely different proof techniques which we feel are more suitable for attacking future research questions in this area. Finally, we complete the classification of two dimensional $S$-free convex sets when $S$ is the intersection of a translated lattice and a polyhedron, thus settling the covering question in two dimensions for such $S$.