When several phases coexist, the interface between two phases can be wetted by several films of the other phases. This is called multilayer wetting and can be characterized by the behavior of the spreading coefficients, which relate the surface tensions between the different phases. In this paper we consider a class of models which can exhibit a sequence of phase transitions, With some new correlation inequalities, we prove the positivity of a family of spreading coefficients. These inequalities, together with a thermodynamic argument, lead to the conclusion of multilayer wetting. These results generalize earlier results where single-layer interfacial wetting was obtained for the Ports model.