High-precision eclipse spectrophotometry of transiting terrestrial exoplanets represents a promising path for the first atmospheric characterizations of habitable worlds and the search for life outside our solar system. The detection of terrestrial planets transiting nearby late-type M-dwarfs could make this approach applicable within the next decade, with soon-to-come general facilities. In this context, we previously identified GJ 1214 as a high-priority target for a transit search, as the transit probability of a habitable planet orbiting this nearby M4.5 dwarf would be significantly enhanced by the transiting nature of GJ 1214 b, the super-Earth already known to orbit the star. Based on this observation, we have set up an ambitious high-precision photometric monitoring of GJ 1214 with the Spitzer Space Telescope to probe the inner part of its habitable zone in search of a transiting planet as small as Mars. We present here the results of this transit search. Unfortunately, we did not detect any other transiting planets. Assuming that GJ 1214 hosts a habitable planet larger than Mars that has an orbital period smaller than 20.9 days, our global analysis of the whole Spitzer dataset leads to an a posteriori no-transit probability of ~98%. Our analysis allows us to significantly improve the characterization of GJ 1214 b, to measure its occultation depth to be 70 ± 35 ppm at 4.5 μm, and to constrain it to be smaller than 205 ppm (3σ upper limit) at 3.6 μm. In agreement with the many transmission measurements published so far for GJ 1214 b, these emission measurements are consistent with both a metal-rich and a cloudy hydrogen-rich atmosphere.