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In a generic spacetime a massless field propagates not just on the surface of the forward lightcone of a source, but in its interior. This inside-the-lightcone "tail radiation" is often described as having "scattered" off the spacetime curvature. In this work, we study the propagation of such tail radiation for a compact, static, spherically symmetric weak-field (i.e. low density) mass distribution that is well off the line-of-sight (LOS) between a source and an observer, and that is coupled to the radiation only gravitationally. For such perturbers, there are four distinct epochs in the observed radiation: the light-cone piece; the subsequent early-time tail -- ending at the first time that a signal moving at the speed of light could travel from the source to a point in the perturber thence to the observer; the subsequent middle-time tail; and the late-time tail, beginning at the last time that a signal could make such a journey. For massless scalar and vector (eg. electromagnetic radiation), we revisit the previously studied early and late-time tail, and perform the first full examination of the middle-time tail. Studying shorter wavelengths and generic perturbers well off the LOS, we find that the late-time tail carries a small fraction of the energy received by the observer; however, the total middle-time tail contains much more energy. We also note that whereas the middle-time tail appears to the observer to emanate from the perturber -- as one might expect for radiation "scattered" from the gravitational perturbation -- the late-time tail appears to come back from the source. We speculate on the potential utility of this middle-time tail for detecting or probing a wide variety of perturbations to the spacetime geometry including dark matter candidates and dark matter halos.