The rod-shaped fission yeast Schizosaccharomyces pombe, which undergoes cycles of monopolar-to-bipolar tip growth, is an attractive organism for studying cell-cycle regulation of polarity establishment. While previous research has described factors mediating this process from interphase cell tips, we found that division site signaling also impacts the re-establishment of bipolar cell growth in the ensuing cell cycle. Complete loss or targeted disruption of the non-essential cytokinesis protein Fic1 at the division site, but not at interphase cell tips, resulted in many cells failing to grow at new ends created by cell division. This appeared due to faulty disassembly and abnormal persistence of the cell division machinery at new ends of fic1Δ cells. Moreover, additional mutants defective in the final stages of cytokinesis exhibited analogous growth polarity defects, supporting that robust completion of cell division contributes to new end-growth competency. To test this model, we genetically manipulated S. pombe cells to undergo new end take-off immediately after cell division. Intriguingly, such cells elongated constitutively at new ends unless cytokinesis was perturbed. Thus, cell division imposes constraints that partially override positive controls on growth. We posit that such constraints facilitate invasive fungal growth, as cytokinesis mutants displaying bipolar growth defects formed numerous pseudohyphae. Collectively, these data highlight a role for previous cell cycles in defining a cell's capacity to polarize at specific sites, and they additionally provide insight into how a unicellular yeast can transition into a quasi-multicellular state.