A temperature-induced commensurate solid-solid phase transition in self-assembled monolayers (SAMs) of alkylthiolates lying on Pt(111) is predicted from molecular dynamics simulations based on ab initio potential energy surfaces. As the system cools down from room temperature to low enough temperature, SAMs of alkylthiolates with more than ~12 carbon atoms undergo an abrupt change of orientation from a nearly upright to a tilted configuration. As the initial hexagonal arrangement of the sulfur headgroups is kept fixed during the simulations, the phase transition is entirely governed by chain-chain interactions. Similar commensurate phase transitions are predicted for hexagonally arranged SAMs with lattice spacings of the order of 4.7-4.9 Å, which, among others, excludes the well-known cases of densely packed SAMs of alkylthiolates on Au(111) and Ag(111). These findings could be relevant for the design of novel electronic or optical devices controllable by temperature.