If coupled feebly to the Standard Model bath, a dark matter can evade the severe constraints from the direct search experiments. At the same time, such interactions help produce dark matter via the freeze-in mechanism. The freeze-in scenario becomes more interesting if one also includes the thermal masses of the different particles involved in the dark matter phenomenology. Incorporating such thermal corrections opens up the possibility of dark matter production via forbidden channels that remain kinematically disallowed in the standard freeze-in setup. Motivated by this, we investigate such freeze-in production of the dark matter in a minimally extended U (1) Lµ−Lτ framework that remains consistent with the recent muon (g − 2) data. Here, the role of the dark matter is played by the scalar with a non-trivial charge under the additional symmetry U (1) Lµ−Lτ . This scalar dark matter obtains a thermally corrected mass at high temperatures for a not-so-small self-coupling. We show that the thermal correction to the dark matter mass plays a significant role in the dark matter phenomenology.