We investigate the Hartman saturation effect of photons traveling through barriers created by stop bands of multilayer stacks. We provide a theoretical basis for the recently observed drastic changes in photon transversal times due to adding a single layer. Using a comprehensive coupled mode analysis and rigorous transfer matrix approach, we show that the experimentally observed effect is caused by surface modes. We investigate the origin and properties of the surface modes and identify them as Tamm-type states.