  Recently, the BESIII Collaboration published the Born cross section of
$e^+e^- \to K^+K^- J/\psi$, in which a new vector enhancement structure
$Y(4500)$ was observed for the first time. The mass of the $Y(4500)$ resonance
structure is in good agreement with our previous prediction for a $5S$-$4D$
mixing charmonium candidate $\psi(4500)$, while the measured width is $2\sigma$
larger than the theoretical estimate. In this work, we reanalyzed the cross
section of $e^+e^- \to K^+K^- J/\psi$ by introducing two theoretically
predicted charmonium states $\psi(4220)$ and $\psi(4500)$ as intermediate
resonances, and found that this width deviation problem can be resolved.
Meanwhile, we found that the inclusion of two established charmonium states
$\psi(4160)$ and $\psi(4415)$ can further improve the fit quality, with four
sets of widely different solutions to $\Gamma_{e^+e^-} \mathcal{B}(\psi \to
K^+K^- J/\psi)$. In order to better identify the properties of the $Y(4500)$,
we calculated the charmonium hadronic transition to final states $K^+K^-
J/\psi$ within the charmed meson loop mechanism, where the fitted branching
ratio of $\psi(4500) \to K^+K^- J/\psi$ can be well explained by the
theoretical calculation, further supporting the new enhancement structure
around 4.5 GeV as a higher charmonium $\psi(4500)$ that we predicted. Finally,
we also discussed the possible signal of another predicted charmonium candidate
$\psi(4380)$ in $e^+e^- \to K^+K^- J/\psi$. These research results should be an
important step in constructing the charmonium family and thoroughly solving the
charmoniumlike $Y$ problem.
