Cells select from a diverse repertoire of migration strategies. Recent developments in tunable biomaterials have helped identify how extracellular matrix properties influence migration, however, many settings lack the fibrous architecture characteristic of native tissues. To investigate migration in fibrous contexts, we independently varied the alignment and stiffness of synthetic 3D fiber matrices and identified two phenotypically distinct migration modes. In contrast to stiff matrices where cells migrated continuously in a traditional mesenchymal fashion, cells in deformable matrices stretched matrix fibers to store elastic energy; subsequent adhesion failure triggered sudden matrix recoil and rapid cell translocation. Across a variety of cell types, traction force measurements revealed a relationship between cell contractility and the matrix stiffness where this migration mode occurred optimally. Given the prevalence of fibrous tissues, an understanding of how matrix structure and mechanics influences migration could improve strategies to recruit repair cells to wound sites or inhibit cancer metastasis.