The slowest Liouvillian mode often bottlenecks the relaxation of an open quantum system to its steady state. Standard strategies circumvent this bottleneck by selecting special initial states or engineering the dissipator. Here we investigate this problem when the input and the subsequent dissipative generator are prescribed. We introduce a pre-dissipative geometric steering protocol that reshapes a prescribed pure or mixed state before relaxation begins—coherent rotations interleaved with nonselective projective measurements—at a fixed Lindblad generator. By steering the state’s Bloch direction along geodesic paths, the protocol suppresses its overlap with the slowest Liouvillian modes. The prepared state can then start farther from equilibrium yet relax faster, realizing the quantum Mpemba effect, as diagnosed by reduced slow-mode overlap and a larger initial distance to stationarity, with distance crossings for complex slow modes verified from the full dynamics. Our framework treats real and complex spectral gaps within a common mode-overlap description. We also quantify the state-side cost of this preparation through cumulative measurement-induced mixing and illustrate its tradeoff with fidelity for a mixed target. We demonstrate Mpemba acceleration in driven qubit and multiqubit systems and discuss the required operations in trapped-ion and superconducting platforms.