  We work with a class of scalar extended theory of gravity that can drive the
present cosmic acceleration as well as accommodate a mild cosmic variation of
the fine structure constant $\alpha$. The motivation comes from a vintage
theory developed by Bekenstein, Sandvik, Barrow and Magueijo. The $\alpha$
variation is introduced by a real scalar field interacting with charged matter.
We execute a cosmological reconstruction based on a parametrization of the
present matter density of the Universe. Observational consistency is ensured by
comparing the theoretical estimates with JLA + OHD + BAO data sets, using a
Markov chain Monte Carlo simulation. An analysis of molecular absorption lines
from HIRES and UVES spectrographs is considered as a reference for the
variation of $\alpha$ at different redshifts. Two examples are discussed. The
first explores a field-dependent kinetic coupling of the scalar field
interacting with charged matter. The second example is a generalized
Brans-Dicke formalism where the varying $\alpha$ is fitted in as an effective
matter field. This generates a simultaneous variation of the Newtonian constant
$G$ and $\alpha$. The pattern of this variation can have a crucial role in
cosmic expansion history.
