Cancer cells are usually found to be softer than normal cells. But their stiffness changes when they are in contact with different environments, due to mechanosensitivity. For example, they adhere to a given substrate by tuning their cytoskeleton, thus affecting their rheological properties. This mechanism could become efficient when cancer cells invade the surrounding tissues, and they have to remodel their cytoskeleton in order to achieve particular deformations. Here we use an atomic force microscope (AFM) in force modulation mode to study how local rheological properties of cancer cells are affected by a change of the environment. Cancer cells are plated on functionalized polyacrylamide substrates of different stiffnesses, as well as on an endothelium substrate. A new correction of Hertz model is developed because measurements require to account for the precise properties of the layered, thin viscoelastic substrates. The main results show the influence of local cell rheology (nucleus, perinuclear, edge locations), and the role of invasiveness. A general mechanosensitive trend is found where the cell elastic modulus and transition frequency increase with substrate elasticity, but this tendency breaks down with a real endothelium substrate. These effects are investigated further during cell transmigration, when the actin cytoskeleton undergoes a rapid reorganization process necessary to push through the endothelial gap, in agreement with the local viscoelastic changes measured by AFM. Taken together, these results introduce a paradigm for a new possible extravasation mechanism.