  Multi-component dark matter particles may have a more intricate direct
detection signal than simple elastic scattering on nuclei. In a broad class of
well-motivated models the inelastic excitation of dark matter particles is
followed by de-excitation via $\gamma$-decay. In experiments with fine energy
resolution, such as many $0\nu 2\beta$ decay experiments, this motivates a
highly model-independent search for the sidereal daily modulation of an
unexpected $\gamma$ line. Such a signal arises from two-step WIMP interaction:
the WIMP is first excited in the lead shielding and subsequently decays back to
the ground state via the emission of a monochromatic $\gamma$ within the
detector volume. We explore this idea in detail by considering the model of
magnetic inelastic WIMPs, and take a sequence of CUORE-type detectors as an
example. We find that under reasonable assumptions about detector performance
it is possible to efficiently explore mass splittings of up to few hundreds of
keV for a WIMP of weak-scale mass and transitional magnetic moments. The
modulation can be cheaply and easily enhanced by the presence of additional
asymmetric lead shielding. We devise a toy simulation to show that a specially
designed asymmetric shielding may result in up to $30\%$ diurnal modulations of
the two-step WIMP signal, leading to additional strong gains in sensitivity.
