Power budgeting techniques allow thermally safe operation in multi-/many-core processors while still allowing for efficient exploitation of available thermal headroom. Core-level power budgeting techniques like Thermal Safe Power (TSP) have allowed for more efficient operations than chip-level power budgeting techniques like Thermal Design Power (TDP) since the finer granularity permits operations closer to the threshold temperature without thermal violations. State-of-the-art TSP bases its power budgeting calculations on the long-term steady-state temperature of cores while ignoring trends in their short-term transient temperature. In this paper, we propose a new power budgeting technique called TTSP (Transient-Temperature-based Safe Power) that bases its calculation on the current temperature of the core, a detail ignored by TSP. T-TSP provides a dynamic power budget to a core, which inversely correlates with the core’s thermal headroom. Dynamic power budgeting with T-TSP allows cores to reach the threshold temperature faster than TSP and operate safely close to it in perpetuity. Therefore, it provides the same thermal guarantees as TSP but enables even more efficient exploitation of thermal headroom. We integrate T-TSP with a state-of-the-art thermal interval simulation toolchain. Our detailed evaluations show that benchmarks execute faster by up to 17.94% and 8.37% on average when we do power budgeting with T-TSP instead of the state-ofthe-art TSP. Finally, we make T-TSP publicly available in both its integrated and stand-alone forms.