The Kohn-Sham exchange potential is shown to approach on nodal surfaces of the energetically highest occupied orbital different asymptotic limits than in other regions. This leads to strong anisotropies and barrier-well structures in the near-asymptotic region. Effective exact-exchange potentials, like the one obtained in the recently introduced localized Hartree-Fock method, exhibit the correct asymptotic limits and the accompanying structures in the near-asymptotic region. An efficient, accurate method for the calculation of the Slater potential and of effective exact-exchange potentials in the asymptotic region is presented. The method is based on an asymptotic continuation of Kohn-Sham (KS) orbitals and is numerically stable up to arbitrary large distances from the molecule. It can easily be implemented in existing quantum chemistry codes employing Gaussian basis sets. The asymptotic region of effective exact-exchange potentials is shown to be different from the various asymptotically corrected exchange-correlation functionals and to strongly affect unoccupied KS orbitals. Results for small- and medium-size molecules are presented.