We extend the method of Simpson and Visser (SV) of regularising a black hole spacetime, to cases where the initial metric represents a globally naked singularity. We choose two particular geometries, the Janis-Newman-Winicour (JNW) metric representing the solution of an Einstein-scalar field system, and the Joshi-Malafarina-Narayan (JMN) metric that represents the asymptotic equilibrium configuration of a collapsing star supported by tangential pressures as the starting configuration. We illustrate several novel features for the modified versions of the JNW and JMN spacetimes. In particular we show that, depending on the values of the parameters involved, the modified JNW metric may represent either a two way traversable wormhole, or it may remain the original naked singularity. On the other hand, the SV modified JMN geometry is always a wormhole. Particle motion and observational aspects of these new geometries are investigated and are shown to possess interesting features.