Topology optimization with polygonal meshes is promising since checkerboards, point-flexures, layering and islanding like singularities get circumvented by the natural imposition of the geometric, 'edge-connectivity' constraint. However, numerous notches get retained on the boundaries of optimal topologies obtained from polygonal tessellations. Previous efforts on Material Mask Overlay Strategy (MMOS) that used hexagonal cells and negative masks have either ignored boundary smoothing, have used it as a post processing step, or have implemented it between the gradient and stochastic searches. Here, we embed boundary smoothing within each iteration of gradient search permitting true evaluation of the objective and the associated sensitivities for all intermediate topologies. Smoothing is performed in a number of steps (represented by parameter β) by systematically shifting the nodes at the boundaries of the continuum. Consequently, some hexagonal cells get degenerated which necessitates their remodeling into Wachspress pentagonal or quadrilateral finite elements to avoid singularity of the stiffness matrix. Material assignment to each cell is accomplished using the logistic function with high values of the material parameter, α approximating the Heaviside function to yield close to binary solutions. However, initial use of high material parameter destabilizes the MMOS since the design sensitivities approach to zero. For stability, α is increased gradually from 1 to an a priori specified value αs. Compared to its predecessors, the modified algorithm shows promise in terms of quality of solutions obtained in least possible number of function evaluations.