  We investigate double-strangeness exchange reactions, $K^-p\to K^+\Xi^-$ and
$K^-p\to K^0\Xi^0$, using an effective Lagrangian approach based on a hybrid
Regge-plus-resonance model involving rescattering diagrams. We consider the
background processes that include $\Lambda$, $\Sigma$, and $\Sigma(1385)$ Regge
trajectories in the $u$ channel and the $s$-channel Born-term diagrams. The
$s$-channel hyperon resonances account for the bump structures in the total
cross sections. The $s$-channel $\Lambda(2100,7/2^-)$ and $\Sigma(2030,7/2^+)$
resonances contribute significantly, and the $\Sigma(2250)$ resonance favors
$J^P = 7/2^-$. The interference pattern between the $\Lambda(2100,7/2^-)$ and
$\Sigma(2030,7/2^+)$ amplitudes is essential for describing forward
differential cross sections in the resonance region $2.07 < \sqrt{s} < 2.15$
GeV. The absence of the $S=-2$ meson leads to the inclusion of the meson-baryon
rescattering diagrams to describe the $K^- p \to K\Xi$ reaction data, proving
the significant contribution of the $(\phi,\rho,\omega)$--$(\Lambda,\Sigma)$
rescattering diagrams.
