The cubic perovskite SrFeO3 was recently reported to host hedgehog-and skyrmion-lattice phases in a highly symmetric crystal structure which does not support the Dzyaloshinskii-Moriya interactions commonly invoked to explain such magnetic order. Hints of a complex magnetic phase diagram have also recently been found in powder samples of the single-layer Ruddlesden-Popper analog Sr2FeO4, so a reinvestigation of the bilayer material Sr3Fe2O7, believed to be a simple helimagnet, is called for. Our magnetization and dilatometry studies reveal a rich magnetic phase diagram with at least six distinct magnetically ordered phases and strong similarities to that of SrFeO3. In particular, at least one phase is apparently multiple-q, and the qs are not observed to vary among the phases. Since Sr3Fe2O7 has only two possible orientations for its propagation vector, some of the phases are likely exotic multiple-q order, and it is possible to fully detwin all phases and more readily access their exotic physics.