Given the ever-increasing role of data centers in global energy consumption, the problem of assigning jobs (items) to servers (bins) in an energy-efficient manner has become more and more important in recent years. In that respect, the temporal bin packing problem with fire-ups (TBPP-FU) takes into account not only the number of servers used but also the switch-on processes required to operate them. Due to the challenging size of the resulting ILP models, various tailored reduction strategies can be applied to obtain more tractable formulations. In this article, we show how the information from a heuristic solution can be used to further improve these exact approaches, extending a theoretical result that was previously proven only for some very restrictive cases in the literature. Finally, the benefits of this new reduction procedure will be demonstrated based on computational tests.