<p>Rapid mass vaccination is critical for curbing transmission during large-scale outbreaks, yet many optimization models emphasize cost or equity without explicitly shaping rollout speed. This study develops a deterministic time-oriented mathematical programming framework for national vaccine distribution that integrates multi-echelon flows, inter-echelon lead times, cold-chain capacity limits, and clinic service constraints. A time-weighted allocation objective is proposed to prioritize earlier coverage and is benchmarked against direct makespan minimization. Results show that the time-weighted formulation consistently increases early cumulative coverage while matching the final completion period obtained under makespan minimization across the tested benchmark instances, yielding a more favorable rollout trajectory without compromising completion time. Scenario experiments indicate that omitting supply availability, lead times, or clinic capacity can substantially underestimate rollout duration. In the Indonesia-based case study, vaccine availability, delivery delays, and vaccination throughput emerge as the dominant drivers of temporal performance, whereas expanding storage capacity provides limited marginal gains once feasibility is ensured. The findings highlight high-leverage interventions that strengthen upstream flow reliability and improve clinical throughput to accelerate population coverage under binding operational constraints.</p>