Piezoelectrics interconvert mechanical energy and electric charge and are widely used in actuators and sensors. The best performing materials are ferroelectrics at a morphotropic phase boundary (MPB), where several phases can intimately coexist. Switching between these phases by electric field produces a large electromechanical response. In the ferroelectric BiFeO$_3$, strain can be used to create an MPB-like phase mixture and thus to generate large electric field dependent strains. However, this enhanced response occurs at localized, randomly positioned regions of the film, which potentially complicates nanodevice design. Here, we use epitaxial strain and orientation engineering in tandem - anisotropic epitaxy - to craft a hitherto unavailable low-symmetry phase of BiFeO$_3$ which acts as a structural bridge between the rhombohedral-like and tetragonal-like polymorphs. Interferometric displacement sensor measurements and first-principle calculations reveal that under external electric bias, this phase undergoes a transition to the tetragonal-like polymorph, generating a piezoelectric response enhanced by over 200%, and associated giant field-induced reversible strain. These results offer a new route to engineer giant electromechanical properties in thin films, with broader perspectives for other functional oxide systems.