  Inspired by the recent measurement of the process $e^+e^-\rightarrow
\Lambda\bar{\Lambda}$, we calculate the mass spectrum of the $\phi$ meson with
the GI model. For the excited vector strangeonium states $\phi(3S,~4S,~5S,~6S)$
and $\phi(2D,~3D,~4D,~5D)$, we further investigate the electronic decay width
with the Van Royen-Weisskopf formula, and the partial widths of the
$\Lambda\bar{\Lambda}$, $\Xi^{-(*)}\bar{\Xi}^+$, and
$\Sigma^{+(*)}\bar{\Sigma}^{-(*)}$ decay modes with the extended quark pair
creation model. We find that the electronic decay width of the $D$-wave vector
strangeonium is about $3\sim8$ times larger than that of the $S$-wave vector
strangeonium. Around 2232 MeV the partial decay width of the
$\Lambda\bar{\Lambda}$ mode can reach up to several MeV for $\phi(3^3S_1)$,
while the partial $\Lambda\bar{\Lambda}$ decay width of $\phi(2^3D_1)$ is
$\mathcal{O}(10^{-3})$ keV. If the threshold enhancement reported by the BESIII
Collaboration arises from the strangeonium meson, this state is very likely to
be the $\phi(3^3S_1)$ state. We also note that the $\Lambda\bar{\Lambda}$ and
$\Sigma^{+}\bar{\Sigma}^{-}$ partial decay widths of the states $\phi(3^3D_1)$
and $\phi(4^3S_1)$ are about several MeV, respectively, which are enough to be
observed in future experiments.
