  Inspired by the observation of the doubly charmed tetraquark state $T_{cc}^+$
at the LHCb Collaboration, we systematically study the $S$-wave interactions
between a pair of charmed mesons ($D_{1},\,D_{2}^{\ast}$) in the $T$-doublet,
where we adopt the one-boson-exchange model and consider both the $S$-$D$ wave
mixing and coupled channel effects. Our numerical results suggest that the
prime doubly charmed molecular tetraquark candidates are the $S$-wave $D_1D_1$
state with $I(J^P)=0(1^+)$, the $S$-wave $D_1{D}_2^{\ast}$ states with
$I(J^P)=0(1^+,\,2^+,\,3^+)$, and the $S$-wave ${D}_2^{\ast}{D}_2^{\ast}$ states
with $I(J^P)=0(1^+,\,3^+)$. The $S$-wave $D_1D_1$ state with $I(J^P)=1(2^+)$,
the $S$-wave $D_1{D}_2^{\ast}$ state with $I(J^P)=1(3^+)$, and the $S$-wave
${D}_2^{\ast}{D}_2^{\ast}$ state with $I(J^P)=1(4^+)$ may be the possible
isovector doubly charmed molecular tetraquark candidates. We expect the
experiments like the LHCb or Belle II collaborations to search for these
predicted doubly charmed molecular tetraquarks near the $D_1D_1$, $D_1D_2^*$,
and $D_2^*D_2^*$ thresholds in future.
