We demonstrate, for the first time, that a quasi-one-dimensional Wigner crystal formed on superfluid (4)He with only a few electrons in the confined direction shows reentrant melting. By transport measurements, we find oscillations in current as a function of linear density measured at a fixed driving voltage at high temperatures, and detailed analyses of transport data reveal that the oscillations originate from the periodic reduction of the melting temperature as a function of linear density. Comparison with the structural phase diagram suggests that the reduction of the melting temperature occurs at the boundaries between the different structures as the structure evolves from a single, double, followed by a triple chain.