Post-disaster emergency logistics is often organized according to prespecified response periods, although operational relief tasks may be completed earlier or later than those calendar boundaries. This study develops a task-completion-triggered multi-stage resource-allocation mechanism (TCMS) for a two-echelon emergency logistics network. Effective first-contact coverage and cumulative forward deployment generate three sequential operational stages—rapid coverage, intensive deployment, and recovery support—through a one-period state-update rule. The operational stage determines the priority regime used to select among feasible allocation schedules. The mechanism is evaluated within a common direct-decision rolling-horizon mixed-integer linear programming (MILP) framework. At each decision epoch, policy-neutral cost-service candidates are generated using total cost and cumulative raw weighted backlog, whereas the current operational stage supplies stage-dependent priority coefficients for selecting the implemented action. TCMS and a fixed-stage baseline therefore differ in the source of their operational stage, while a stage-free rolling-horizon benchmark retains a neutral priority regime. Computational experiments include formulation validation, Base and Medium rolling-horizon policy comparisons, a Large static MILP reference, and trigger-parameter and stage-priority-coefficient sensitivity analyses. The results show that endogenous stage timing changes the selected cost-service operating point, with effects varying across instances. Sensitivity analysis further shows that transition timing and allocation outcomes can vary with trigger settings and stage-priority calibration. TCMS is particularly suited to settings in which transition timing cannot be specified reliably in advance, and managers require an observable basis for adjusting allocation priorities as relief operations progress.