  The space-borne missions CoRoT and Kepler have already brought stringent
constraints on the internal structure of low-mass evolved stars, a large part
of which results from the detection of mixed modes. However, all the potential
of these oscillation modes as a diagnosis of the stellar interior has not been
fully exploited yet. In particular, the coupling factor or the gravity-offset
of mixed modes, $q$ and $\varepsilon_{\rm g}$, are expected to provide
additional constraints on the mid-layers of red giants, which are located
between the hydrogen-burning shell and the neighborhood of the base of the
convective zone. In the present paper, we investigate the potential of the
coupling factor in probing the mid-layer structure of evolved stars. Guided by
typical stellar models and general physical considerations, we modeled the
coupling region along with evolution. We subsequently obtained an analytical
expression of $q$ based on the asymptotic theory of mixed modes and compared it
to observations. We show that the value of $q$ is degenerate with respect to
the thickness of the coupling evanescent region and the local density scale
height. A structural interpretation of the global variations in $q$ observed on
the subgiant and the red giant branches, as well as on the red clump, was
obtained in the light of this model. We demonstrate that $q$ has the promising
potential to probe the migration of the base of the convective region as well
as convective extra-mixing in evolved red giant stars with typically $\nu_{\rm
max} \lesssim 100~\mu$Hz. We also show that the frequency-dependence of $q$
cannot be neglected in the oscillation spectra of such stars, which is in
contrast with what is assumed in the current measurement methods. This
analytical study paves the way for a more quantitative exploration of the link
of $q$ with the internal properties of evolved stars using stellar models.
