  Recently the LHCb collaboration has announced the discovery of the $T_{cc}^+$
tetraquark. Being merely a few hundred ${\rm keV}$ below the $D^{*+} D^0$
threshold, the $T_{cc}^+$ is expected to have a molecular component, for which
there is a good separation of scales that can be exploited to make reasonably
accurate theoretical predictions about this tetraquark. Independently of its
nature, the most important decay channels will be $D^+ D^0 \pi^0$, $D^0 D^0
\pi^+$ and $D^+ D^0 \gamma$. Its closeness to threshold suggests that the mass
and particularly the width of the $T_{cc}^+$ tetraquark depend on the resonance
profile. While the standard Breit-Wigner parametrization generates a $T_{cc}^+$
that is too broad for current theoretical calculations to reproduce, a
three-body unitarized Breit-Wigner shape reveals instead a decay width
($\Gamma_{\rm pole} = 48\pm 2\,{}^{+0}_{-12}\,{\rm keV}$) consistent with
theoretical expectations. Here we consider subleading order contributions to
the decay amplitude, which though having at most a moderate impact in the width
still indicate potentially significant differences with the experimental width
that can be exploited to disentangle the nature of the $T_{cc}^+$. Concrete
calculations yield $\Gamma^{\rm LO} = 49 \pm 16\,{\rm keV}$ and $\Gamma^{\rm
NLO} = 58^{+7}_{-6}\,{\rm keV}$, though we expect further corrections to the
${\rm NLO}$ decay widths from asymptotic normalization effects. We find that a
detailed comparison of the ${\rm NLO}$ total and partial decay widths with
experiment suggests the existence of a small (but distinguishable from zero)
non-molecular component of the $T_{cc}^+$.
