We report on the search for weakly interacting massive particle (WIMP) dark matter candidates in the galactic halo that interact with sodium and iodine nuclei in the COSINE-100 experiment and produce energetic electrons that accompany recoil nuclei via the Migdal effect. The WIMP mass sensitivity of previous COSINE-100 searches that relied on the detection of ionization signals produced by target nuclei recoiling from elastic WIMP-nucleus scattering was restricted to WIMP masses above ∼5 GeV/c2 by the detectors' 1 keVee energy-electron-equivalent threshold. The search reported here looks for recoil signals enhanced by the Migdal electrons that are ejected during the scattering process. This is particularly effective for the detection of low-mass WIMP scattering from the crystals' sodium nuclei in which a relatively larger fraction of the WIMP's energy is transferred to the nucleus recoil energy and the excitation of its orbital electrons. In this analysis, the low-mass WIMP search window of the COSINE-100 experiment is extended to WIMP mass down to 200 MeV/c2. The low-mass WIMP sensitivity will be further improved by lowering the analysis threshold based on a multivariable analysis technique. We consider the influence of these improvements and recent developments in detector performance to reevaluate sensitivities for the future COSINE-200 experiment. With a 0.2 keVee analysis threshold and high light-yield NaI(Tl) detectors (22 photoelectrons/keVee), the COSINE-200 experiment can explore low-mass WIMPs down to 20 MeV/c2 and probe previously unexplored regions of parameter space.