Quantum coupling in arrayed nanostructures can produce novel mesoscale properties such as electronic minibands to improve the performance of optoelectronic devices, including ultra-efficient solar cells and infrared photodetectors. Colloidal PbSe quantum dots (QDs) that self-assemble into epitaxially-fused superlattices (epi-SLs) are predicted to exhibit such collective phenomena. Here, we show the emergence of distinct local electronic states induced by crystalline necks that connect individual PbSe QDs and modulate the bandgap energy across the epi-SL. Multi-probe scanning tunneling spectroscopy shows bandgap modulation from 0.7 eV in the QDs to 1.1 eV at their necks. Complementary monochromated electron energy-loss spectroscopy demonstrates bandgap modulation in spectral mapping, confirming the presence of these distinct energy states from necking. The results show the modification of the electronic structure of a precision-made nanoscale superlattice, which may be leveraged in new optoelectronic applications.