Context. The power of asteroseismology relies on the capability of global oscillations to infer the stellar structure. For evolved stars, we benefit from unique information directly carried out by mixed modes that probe their radiative cores. This third article of the series devoted to mixed modes in red giants focuses on their coupling factors, which have remained largely unexploited up to now. Aims. With the measurement of coupling factors, we intend to give physical constraints on the regions surrounding the radiative core and the hydrogen-burning shell of subgiants and red giants. Methods. A new method for measuring the coupling factor of mixed modes was implemented, which was derived from the method recently implemented for measuring period spacings. This new method was automated so that it could be applied to a large sample of stars. Results. Coupling factors of mixed modes were measured for thousands of red giants. They show specific variation with mass and evolutionary stage. Weak coupling is observed for the most evolved stars on the red giant branch only; large coupling factors are measured at the transition between subgiants and red giants as well as in the red clump. Conclusions. The measurement of coupling factors in dipole mixed modes provides a new insight into the inner interior structure of evolved stars. While the large frequency separation and the asymptotic period spacings probe the envelope and core, respectively, the coupling factor is directly sensitive to the intermediate region in between and helps determine its extent. Observationally, the determination of the coupling factor is a prior to precise fits of the mixed-mode pattern and can now be used to address further properties of the mixed-mode pattern, such as the signature of buoyancy glitches and core rotation.