Diffusion of small molecules in amorphous polymers is known to follow a form of so-called hopping motion: penetrant molecules are trapped in microscopic cavities for extended time periods; diffusion is made possible by rare but fast jumps between neighboring cavities. Existing understanding of the hopping mechanism is based on the inspection of molecular images during individual molecular-dynamics trajectories. We focus on the diffusion of water molecules in a hydrophilic polymer below its glass transition temperature. The transition path ensemble of one hopping event is sampled with aimless shooting, a type of transition path sampling technique. In these trajectories, configurations of both the penetrant and the polymer change during the transition. Statistical analysis of the ensemble using likelihood maximization leads to a reaction coordinate of the transition, whose key components include the penetrant configuration and distances between penetrant host atom pairs that have strong electrostatic interactions. Polymer motions do not contribute directly to the reaction coordinate. This result points toward a transition mechanism dominated by the penetrant movement. Molecular insights from this study can benefit the development of computational tools that better predict material transport properties, facilitating the design of new materials, including polymers with engineered drying properties.