In a study of $K\overline{K}$ pairs produced in ${\ensuremath{\pi}}^{\ensuremath{-}}p$ interactions from 1.5 to 4.2 BeV/c, we observe the ${{K}_{1}}^{0}{{K}_{1}}^{0}$ threshold enhancement at all beam momenta, and the $\ensuremath{\phi}$ meson at beam momenta below 2.3 BeV/c. There are no significant enhancements in the ${K}^{0}{K}^{\ensuremath{-}}$ system near threshold.