  With the assumptions that the $T_{cc}^+$ discovered at LHCb is a $D^{*}D$
hadronic molecule, using a nonrelativistic effective field theory we calculate
the radiative partial widths of $T_{cc}^* \to D^*D\gamma$ with $T_{cc}^*$ being
a $D^{*}D^{*}$ shallow bound state and the heavy-quark-spin partner of
$T_{cc}^+$. The $I=0$ $D^*D$ rescattering effect with the $T_{cc}$ pole is
taken into account. The results show that the isoscalar $D^{\ast} D$
rescattering can increase the tree-level decay width of $T_{cc}^{\ast
+}\rightarrow D^{*+}D^0\gamma$ by about $50\%$, while decrease that of
$T_{cc}^{\ast +}\rightarrow D^{*0}D^+\gamma$ by a similar amount. The two-body
partial decay widths of the $T_{cc}^{*+}$ into $T_{cc}^+\gamma$ and
$T_{cc}^+\pi^0$ are also calculated, and the results are about $6~\rm{keV}$ and
$3~\rm{keV}$, respectively. Considering that the $D^*$ needs to be
reconstructed from the $D\pi$ or $D\gamma$ final state in an experimental
measurement, the four-body partial widths of the $T_{cc}^{*+}$ into
$DD\gamma\gamma$ and $DD\pi\gamma$ are explicitly calculated, and we find that
the interference effect between different intermediate $D^*D\gamma$ states is
small. The total radiative decay width of the $T_{cc}^*$ is predicted to be
about $24~\rm{keV}$. Adding the hadronic decay widths of $T_{cc}^* \to
D^*D\pi$, the total width of the $T_{cc}^*$ is finally predicted to be
$(65\pm2)$ keV.
