We investigate a three-dimensional kinetically-constrained model that exhibits two types of phase transitions and different densities. At the jamming density \rho_J there is a mixed-order phase transition in which a finite fraction of the particles become frozen, but the other particles may still diffuse throughout the system. By identifying an underlying structure we avoid simulating the entire dynamics of the system. Instead we use a culling algorithm to learn about diffusion in a deterministic fasion. We find that at the caging density \rho_C > \rho_J the mobile particles become trapped in finite cages and no longer diffuse. The caging transition occurs due to a percolation transition of the unfrozen sites, and we numerically find that it is a continuous transition with the same critical exponents as random percolation.