We have used self-consistent field theory (SCFT) simulations to investigate the effect of film thickness and chain architecture on the temperature-dependent sphere to cylinder order-order transition in diblock and triblock copolymers. While the order-order transition temperature (OOT) for bulk phases depends only on the volume fraction for a particular block copolymer, for thin films we find a strong dependence on film thickness and the nature of chains forming the surface wetting layer (i.e., ends vs mid-monomers). Our studies show that subtle differences in the packing frustration of the spherical and cylindrical phases as well as the higher configurational entropy of free chain ends at the surface are sufficient to drive the equilibrium configuration in thin films away from the stable bulk structure. These contributions to the free energy that arise in thin films but are absent in bulk cause the thin film OOT to be shifted from the bulk OOT by Delta chi N similar to 5.