Specific heat, resistivity, magnetic susceptibility, linear thermal expansion (LTE), and high-resolution synchrotron x-ray powder diffraction investigations of single crystals Fe_1+yTe (0.06 ≤y≤ 0.15) reveal a splitting of a single, first-order transition for y≤ 0.11 into two transitions for y ≤ 0.13. Most strikingly, all measurements on identical samples Fe_1.13Te consistently indicate that, upon cooling, the magnetic transition at T_N precedes the first-order structural transition at a lower temperature T_s. The structural transition in turn coincides with a change in the character of the magnetic structure. The LTE measurements along the crystallographic c axis display a small distortion close to T_N due to a lattice striction as a consequence of magnetic ordering, and a much larger change at T_s. The lattice symmetry changes, however, only below T_s as indicated by powder x-ray diffraction. This behavior is in stark contrast to the sequence in which the phase transitions occur in Fe pnictides.