Adhesion-independent migration is a prominent mode of cell motility in confined environments, yet the physical principles that guide such movement remain incompletely understood. We present a phase-field model for simulating the motility of deformable, non-adherent cells driven by contractile surface instabilities of the cell cortex. This model couples surface and bulk hydrodynamics, accommodates large shape deformations and incorporates a diffusible contraction-generating molecule (myosin) that drives cortical flows. These capabilities enable a systematic exploration of how mechanical cues direct cell polarization and migration. We first demonstrate that spontaneous symmetry breaking of cortical activity can lead to persistent and directed movement in channels. We then investigate how various physical cues – including gradients in friction, viscosity, and channel width as well as external flows and hydrodynamic interactions between cells – steer migration. Our results show that active surface dynamics can generate stimulus-specific cell behaviors, such as migration up friction gradients or escape from narrow regions. Beyond cell migration, the model offers a versatile platform for exploring the mechanics of active surfaces in biological systems.