We propose a new $CPT$-even and Lorentz-violating nonminimal coupling between fermions and Abelian gauge fields involving the $CPT$-even tensor $({K}_{F}{)}_{\ensuremath{\mu}\ensuremath{\nu}\ensuremath{\alpha}\ensuremath{\beta}}$ of the standard model extension. We thus investigate its effects on the cross section of the electron-positron scattering by analyzing the process ${e}^{+}+{e}^{\ensuremath{-}}\ensuremath{\rightarrow}{\ensuremath{\mu}}^{+}+{\ensuremath{\mu}}^{\ensuremath{-}}$. Such a study was performed for the parity-odd and parity-even nonbirefringent components of the Lorentz-violating $({K}_{F}{)}_{\ensuremath{\mu}\ensuremath{\nu}\ensuremath{\alpha}\ensuremath{\beta}}$ tensor. Finally, by using experimental data available in the literature, we have imposed upper bounds as tight as ${10}^{\ensuremath{-}12}\text{ }\text{ }(\mathrm{eV}{)}^{\ensuremath{-}1}$ on the magnitude of the $CPT$-even and Lorentz-violating parameters while nonminimally coupled.