The density of Andreev levels in a normal metal (N) in contact with two superconductors (S) is known to exhibit an induced minigap related to the inverse dwell time. We predict a small secondary gap just below the superconducting gap edge - a feature that has been overlooked so far in numerous microscopic studies of the density of states in SNS structures. In a generic structure with N being a chaotic cavity, the secondary gap is the widest at zero phase bias. It closes at some finite phase bias, forming the shape of a "smile". Asymmetric couplings give even richer gap structures near the phase difference π. All the features found should be amendable to experimental detection in high-resolution low-temperature tunneling spectroscopy.