A combined theoretical and experimental investigation has been undertaken to determine optimum conditions for achieving rapid cooling of H2 clusters in nozzle-beam expansions with the goal of producing superfluid H2 clusters. Theory predicts that a temperature less than 6.6 K, well below the 13.8 K triple-point temperature of p-H2, is required. Terminal specific enthalpies of clusters are determined experimentally from terminal velocities of clusters measured using the time-of-flight technique. The results are interpreted in the context of isothermal and adiabatic spinodals for p-H2 constructed using thermodynamic methods and a van der Waals equation-of-state model. The lowest terminal enthalpies are achieved in expansions starting from supercritical source conditions and crossing the binodal curve of the phase diagram with densities far to the liquid side of the critical point. In this case the clusters are formed via relatively late fragmentation of metastable liquid H2. These clusters are expected to be liquid and, at the point in the expansion at which collisions cease, to have temperatures of about 9 K. Before arriving at the detector they are cooled further by evaporation to temperatures between 4 and 5 K. Further experiments are needed to determine if these clusters are superfluid.