In this article we introduce the m-cover poset of an arbitrary bounded poset P, which is a certain subposet of the m-fold direct product of P with itself. Its ground set consists of multichains of P that contain at most three different elements, one of which has to be the least element of P, and the other two elements have to form a cover relation in P. We study the m-cover poset from a structural and topological point of view. In particular, we characterize the posets whose m-cover poset is a lattice for all m>0, and we characterize the special cases, where these lattices are EL-shellable, left-modular, or trim. Subsequently, we investigate the m-cover poset of the Tamari lattice Tn, and we show that the smallest lattice that contains the m-cover poset of Tn is isomorphic to the m-Tamari lattice Tn(m) introduced by Bergeron and Préville-Ratelle. We conclude this article with a conjectural desription of an explicit realization of Tn(m) in terms of m-tuples of Dyck paths.