  We consider warped five-dimensional thick braneworlds with four-dimensional
Poincar\'e invariance originated from bulk scalar matter non-minimally coupled
to gravity plus a Gauss-Bonnet term. The background field equations as well as
the perturbed equations are investigated. A relationship between 4D and 5D
Planck masses is studied in general terms. By imposing finiteness of the 4D
Planck mass and regularity of the geometry, the localization properties of the
tensor modes of the perturbed geometry are analyzed to first order, for a wide
class of solutions. In order to explore the gravity localization properties for
this model, the normalizability condition for the lowest level of the tensor
fluctuations is analyzed. It is found that for the examined class of solutions,
gravity in 4 dimensions is recovered if the curvature invariants are regular
and the 4D Planck mass is finite. It turns out that both the addition of the
Gauss-Bonnet term and the non-minimal coupling between the scalar field and
gravity {\it reduce} the value of the 4D Planck mass compared to its value when
the scalar field and gravity are minimally coupled and the Gauss-Bonnet term is
absent. The above discussed analysis depends on the explicit form of the scalar
field (through its non-minimal coupling to gravity), making necessary the
construction of explicit solutions in order to get results in closed form, and
is illustrated with some examples which constitute smooth generalizations of
the so-called Randall-Sundrum braneworld model. These solutions were obtained
by making use of a detailed {\it singular perturbation theory} procedure with
respect to the non-minimal coupling parameter between the scalar field and
gravity, a difficult task that we managed to perform in such a way that all the
physically meaningful conditions for the localization of gravity are fully
satisfied. From the obtained...
