The magnetic structure of wedge-shaped Co/Au(111) films has been analyzed in the thickness range of around five monolayers where a reorientation transition takes place. The detection of two critical thicknesses with an apparently new phase in between them can be understood by evoking general considerations of stability and coexistence of phases. The analysis in terms of thickness-driven evolution in the anisotropy space of the system leads to the determination of the surface anisotropy constants of first and second order in a most direct and consistent manner.