We parameterize a large class of corrections to general relativity which include a long-ranged gravitational scalar field as a dynamical degree of freedom in two ways: parameterizing the structure of the correction to the action, and parameterizing the scalar hair (multipole structure) that compact objects and black holes attain. The presence of this scalar hair violates the no-hair theorems present in general relativity, which leads to several important effects. The effects we consider are i) the interaction between an isolated body and an external scalar field, ii) the scalar multipole-multipole interaction between two bodies in a compact binary, iii) the additional pericenter precession of a binary, iv) the scalar radiation from a binary, and v) the modification to the gravitational wave phase from a binary. We apply this framework to example theories including Einstein-dilaton-Gauss-Bonnet gravity and dynamical Chern-Simons gravity, and estimate the size of the effects. Finally we estimate the bounds that can be placed on parameters of the theories from the precession of pulsar binaries and from gravitational waves.