  Stimulated by the newly reported doubly-charmed tetraquark state $T_{cc}^+$
by LHCb, we carry out a systematic investigation of the $S$-wave interactions
between the charmed meson ($D,\,D^{*}$) in $H$-doublet and the charmed meson
($D_{1},\,D_{2}^{*}$) in $T$-doublet by adopting the one-boson-exchange model.
Both the $S$-$D$ wave mixing effect and the coupled channel effect are taken
into account. By performing a quantitative calculation, we suggest that the
$S$-wave $D^{*} D_{1}$ states with $I(J^{P})=0(0^{-},\,1^{-})$ and the $S$-wave
$D^{*}D_{2}^{*}$ state with $I(J^{P})=0(1^{-})$ should be viewed as the most
promising candidates of the doubly-charmed molecular tetraquark states, and the
$S$-wave $DD_{1}$ state with $I(J^{P})=0(1^{-})$, the $S$-wave $DD_{2}^{*}$
state with $I(J^{P})=0(2^{-})$, and the $S$-wave $D^{*}D_{2}^{*}$ state with
$I(J^{P})=0(2^{-})$ are the possible doubly-charmed molecular tetraquark
candidates. With the accumulation of experimental data at Run III and after
High-Luminosity-LHC upgrade, these predicted doubly-charmed molecular
tetraquark states can be accessible at LHCb in the near future.
