It has been shown that an inhomogeneously broadened optical transition shaped into an atomic frequency comb can store a large number of temporal modes of the electromagnetic field at the single-photon level without the need to increase the optical depth of the storage material. The readout of light modes is made efficient thanks to the rephasing of the optical-wavelength coherence similar to photon-echo-type techniques, and the reemission time is given by the comb structure. For on-demand readout and long storage times, two control fields are used to transfer the optical coherence back and forth into a spin wave. Here, we present a detailed analysis of the spin-wave storage based on chirped adiabatic control fields. In particular, we verify that chirped fields require significantly weaker intensities than π pulses. The price to pay is a reduction of the multimode storage capacity that we quantify for realistic material parameters associated with solids doped with rare-earth-metal ions.