  We discuss recent lattice data for the $T_{cc}(3875)^+$ state to stress, for
the first time, a potentially strong impact of left-hand cuts from the one-pion
exchange on the pole extraction for near-threshold exotic states. In
particular, if the left-hand cut is located close to the two-particle
threshold, which happens naturally in the $DD^*$ system for the pion mass
exceeding its physical value, the effective-range expansion is valid only in a
very limited energy range up to the cut and as such is of little use to
reliably extract the poles. Then, an accurate extraction of the pole locations
requires the one-pion exchange to be implemented explicitly into the scattering
amplitudes. Our findings are general and potentially relevant for a wide class
of hadronic near-threshold states.
