  Within the framework of dispersion theory, we study the the processes
$e^+e^-\to \phi(2170) \to \phi \pi\pi(K\bar{K})$. The strong pion-pion
final-state interactions, especially the $K\bar{K}$ coupled channel in the
$S$-wave, are taken into account in a model-independent way using the Omn\`es
function solution. Through fitting the experimental data of the $\pi\pi$ and
$\phi\pi$ invariant mass distributions of the $e^+e^- \to \phi(2170) \to \phi
\pi^+\pi^-$ process, the low-energy constants in the chiral Lagrangian are
determined. The theoretical prediction for the cross sections ratio
${\sigma(e^+e^- \to \phi(2170)\to \phi K^+ K^-)}/{\sigma(e^+e^- \to
\phi(2170)\to \phi\pi^+\pi^-)}$ is given, which could be useful for selecting
the physical solution when the fit to the $e^+e^- \to \phi K^+ K^-$ cross
section distribution is available in the future. Our results suggest that above
the kinematical threshold of $\phi K\bar{K}$, the mechanism $e^+e^- \to \phi
K^+ K^-$ with the kaons rescattering to a pion pair plays an important role in
the $e^+e^- \to \phi\pi^+\pi^-$ transition.
