  The $T_{cc}^+$ exotic meson, discovered by the LHCb Collaboration in 2021,
can be interpreted as a molecular state of $D^{(*)0}$ and $D^{(*)+}$ mesons. We
compute next-to-leading-order (NLO) contributions to the strong decay of
$T_{cc}^+$ in an effective field theory for $D$ mesons and pions, considering
contributions from one-pion exchange and final-state rescattering. Corrections
to the total width, as well as the differential distribution in the invariant
mass of the final-state $D$ meson pair are computed. The results remain in good
agreement with LHCb experimental results when the NLO contributions are added.
The leading uncertainties in the calculation come from terms which depend on
the scattering length and effective range in $D$ meson scattering.
