A versatile semiclassical approximation for intense laser-atom processes is presented. This uniform approximation is no more complicated than the frequently used multidimensional saddle-point approximation and far superior, since it applies for all energies, both close to as well as away from the classical cutoffs. In the latter case, it reduces to the standard saddle-point approximation. The uniform approximation agrees accurately with numerical evaluations for potentials, for which these are feasible, and constitutes a practicable method of calculation, in general. The method is applied to the calculation of high-order above-threshold ionization spectra with various binding potentials: Coulomb, Yukawa, and shell potentials which may model C60 molecules or clusters. The shell potentials generate rescattering spectra that are more structured and may feature an apparently higher cutoff.