In this article we present a new perturbation method for hydrogen-like systems with a spinless orbiting particle.We apply the perturbation theory to an exact linearized version of the Klein-Gordon equation obtained by a rescaling of the mass and the energy of the system. For this reason, the relativistic corrections are automatically included. The predictions obtained are compared and tested against an other perturbation approach that uses the Feshbach-Villars formalism and with a self-consistent calculation. We use this new perturbation method to calculate the hyperfine structure of pionic and kaonic atoms providing an additional term in the calculation of the energy transition of these systems. Such a correction is required for a recent measurement of the pion mass.