  Existing data on the $e^+e^-\to K^+K^-$ cross section at the center-of-mass
energy above 2.6 GeV are fitted with a sum of $\psi(3770)$ resonant and
continuum contributions. Two solutions for the resonance production cross
section are found with a significance of 3.2$\sigma$. Data on the $e^+e^-\to
K_SK_L$ cross section are used to resolve the ambiguity and for further
constraining the values of the $e^+e^-\to \psi(3770) \to K^+K^-$ cross section
and the interference phase. They are found to be
$\sigma_{\psi(3770)}=0.073^{+0.061}_{-0.044}$ pb and
$\phi=(309^{+17}_{-35})^\circ$, respectively. The same fitting procedure for
the $\psi(4160)$ resonance leads to the upper limit on the $e^+e^-\to
\psi(4160) \to K^+K^-$ cross section $\sigma_{\psi(4160)}<0.062$ pb at 90\%
confidence level.
