In dynamic flow fields such as winds and ocean currents an agent can navigate by going with the flow, only using minimal propulsion to nudge itself into beneficial flows. This navigation paradigm of hitchhiking flows is highly energy-efficient. However, reliable navigation in this setting remains challenging as typically only forecasts are available which differ significantly from the true currents and the forecast error can be larger than can be handled by the actuation of the agent. In this paper, we propose a novel control method for reliable navigation of underactuated agents hitchhiking flows based on imperfect forecasts. In the spirit of Model Predictive Control our method allows for time-optimal replanning at every time step with only one computation per forecast. Using the recent Multi-Time Hamilton-Jacobi Reachability formulation we obtain a value function which is then used for closed-loop control. We evaluate the reliability of our method empirically over a large set of multi-day start-target missions in the ocean currents of the Gulf of Mexico with realistic forecast errors. Our method outperforms the baselines significantly, achieving high reliability, measured as the success rate of navigating from start to target, at low computational cost.