  We employ a chiral effective field theory-based approach to connect $DD^*$
scattering observables at the physical and variable pion masses accessible in
lattice QCD simulations. We incorporate all relevant scales associated with
three-body $DD\pi$ dynamics and the left-hand cut induced by the one-pion
exchange for pion masses higher than the physical one, as required by
analyticity and unitarity. By adjusting the contact interactions to match
experimental data at the physical pion mass and lattice finite-volume energy
levels at $m_{\pi} = 280$ MeV, we predict the trajectory of the $T_{cc}$ pole
as a function of the pion mass, finding it consistent with the
hadronic-molecule scenario. In particular, we find that the explicit treatment
of the one-pion exchange has a pronounced effect on the pole trajectory for
$m_\pi \gtrsim 230$ MeV by pushing it into the complex energy plane.
