We perform a numerical investigation of the fine splitting of the pseudoscalar meson mass in the Schwinger model. We use overlap fermions at a single lattice spacing and in a mass range, where the analytic prediction $M_{\pi} = 2.008...\; m_f^{2/3}$ is satisfied. We then check the prediction of an exponential suppression of the fine splitting in the fermion mass. We generically find behavior compatible with exponential suppression, but with a prefactor that seems to substantially differ from a leading order expansion.