  Using $980~fb^{-1}$ of data collected on and around the
$\Upsilon(nS)(n=1,2,3,4,5)$ resonances with the Belle detector at the KEKB
collider, we measure the cross section of $e^{+}e^{-}\to \eta\phi$ from
threshold to $3.95~{\rm GeV}/c^{2}$ via initial-state radiation. From a
multi-parameter fit assuming $\phi(2170)$ exists in the $\eta\phi$ final state
according to previous measurement by BESIII, the resonant parameters of
$\phi(1680)$ are determined to be $m_{\phi(1680)} = (1683 \pm 7 \pm 9)~{\rm
MeV}/c^{2}$ (statistical and systematic errors, respectively),
$\Gamma_{\phi(1680)} = (149 \pm 12 \pm 13)~{\rm MeV}$ and, depending on the
possible presence of interfering resonances,
$\Gamma^{e^{+}e^{-}}_{\phi(1680)}\cdot {\cal B}[\phi(1680)\to\eta\phi] = (122
\pm 6 \pm 13)~{\rm eV}$, $(219 \pm 15 \pm 18)~{\rm eV}$, $(163 \pm 11 \pm
13)~{\rm eV}$ or $(203 \pm 12 \pm 18)~{\rm eV}$. The branching fraction of
$\phi(1680)\to\eta\phi$ decay is determined to be approximately 20%.
Additionally, the branching fraction for $J/\psi\to\eta\phi$ is measured to be
$(7.1\pm 1.0 \pm 0.5)\times 10^{-4}$. However, there is no significant observed
$\phi(2170)$ signal in the $\eta\phi$ final states in this analysis, and
correspondingly the upper limit for $\Gamma^{e^{+}e^{-}}_{\phi(2170)}\cdot
{\cal B}(\phi(2170)\to\eta\phi)$ is determined to be either $0.17~{\rm eV}$
(for two fits), or $18.6~{\rm eV}$ (remaining two fits), at 90% confidence
level.
