Solution-processed core/multishell semiconductor quantum dots (QDs) could be tailored to facilitate the carrier separation, promotion, and recombination mechanisms necessary to implement photon upconversion. In contrast to other upconversion schemes, upconverting QDs combine the stability of an inorganic crystalline structure with the spectral tunability afforded by quantum confinement. Nevertheless, their upconversion quantum yield (UCQY) is fairly low. Here, design rules are uncovered that enable to significantly enhance the performance of double QD upconversion systems, and these findings are leveraged to fabricate upconverting QDs with increased photon upconversion efficiency and reduced saturation intensities under pulsed excitation. The role of the intra-QD band alignment is exemplified by comparing the upconversion process in PbS/CdS/ZnSe QDs with that of PbS/CdS/CdSe ones with variable CdSe shell thicknesses. It is shown that electron delocalization into the shell leads to a longer-lived intermediate state in the QDs, facilitating further absorption of photons, and enhancing the upconversion process. The performance of these upconversion QDs under pulsed excitation versus continuous pumping is also compared; the reasons for the significant differences between these two regimes are discussed. The results show how one can overcome some of the limitations of previous upconverting QDs, with potential applications in biophotonics and infrared detection.