  The $e^+e^-\to \omega\eta$ and $e^+e^-\to\omega\pi^0\pi^0$ processes are
ideal platforms to search for higher $\omega$ states. Focusing on the
observations of two enhancement structures around 2.2 GeV existing in
$e^+e^-\to \omega\eta$ and $e^+e^-\to\omega\pi^0\pi^0$ at BESIII, we analyze
how the $\omega(4S)$ and $\omega(3D)$ states play the role in the $e^+e^-\to
\omega\eta$ and $e^+e^-\to\omega\pi^0\pi^0$ processes. The present study is
supported by theoretical $\omega$ mesonic spectroscopy. For reproducing the
data of the cross sections of $e^+e^-\to\omega\eta$ and $\omega\pi^0\pi^0$, the
intermediate $\omega(4S)$ and $\omega(3D)$ should be introduced, which
indicates that the enhancement structures around 2.2 GeV existing in
$e^+e^-\to\omega\eta$ and $\omega\pi^0\pi^0$ contain the $\omega(4S)$ and
$\omega(3D)$ signals. Nonetheless, in the process $e^+e^-\to\omega\eta$, the
$\omega(4S)$ plays a dominant role, while the $\omega(4S)$ and $\omega(3D)$
have similar sizable contributions in the process of
$e^+e^-\to\omega\pi^0\pi^0$, which leads to a difference in the line shape of
enhancement structure in the cross sections under the interference effect.
Thus, we find a solution to alleviate the puzzling difference of resonance
parameter of two reported enhancement structures around 2.2 GeV existing in
$e^+e^-\to \omega\eta$ and $e^+e^-\to\omega\pi^0\pi^0$ at BESIII. The present
study provides valuable information to construct $\omega$ meson family, which
can be accessible at future experiment like BESIII.
