Three alkali metal hydrogen phthalate salts (denoted as XAP, X = K, Rb, and Cs) are chosen as substrates for the growth of quaterthiophene (4T) and hexathiophene (6T) thin films by organic molecular beam epitaxy to study the influence of gradual changes of surface unit cell parameters on the epitaxialgrowth. The increment of substrate lattice parameters increases the distance between the planes that define the furrows where oligothiophene molecules lie, while keeping unmodified the interactions between the overlayer and the substrate. Atom-atom potential simulations predict the preferential azimuthal orientations of the overlayer, which are compared with those experimentally observed. The agreement between simulations and experiments about contact planes and orientation of the crystalline domains in the films is satisfactory for both 4T/XAP and 6T/XAP. The increasing width of the surface furrows existing on moving from KAP to RbAP and CsAP does not cause any significant variation of the orientation and density of the overlayer domains, demonstrating that the interaction between overlayer and substrate is the key factor guiding organic epitaxialgrowth.