We theoretically elucidate the transfer of phonon angular momentum by acoustic modes across a smooth interface between crystals. We analyze this process, which is difficult to describe with the conventional acoustic mismatch model, using a reformulated boundary condition and the Boltzmann theory. For an interface between a chiral and an achiral crystal, our analysis reveals that thermal gradients in the chiral crystal induce angular momentum, which diffuses into the achiral crystal even without heat flow. Notably, within an approximation retaining the chirality-induced population imbalance, the density of angular momentum can be enhanced near the interface. These findings advance our understanding of phonon transport and its interplay with electron spins.