  The spectrograph ESPRESSO recently obtained a limit on the variation of the
fine-structure constant, $\alpha$, through measurements along the line of sight
of a bright quasar with a precision of $1.36$ ppm at $1\sigma$ level. This
imposes new constraints on cosmological models with a varying $\alpha$. We
assume such a model where the electromagnetic sector is coupled to a scalar
field dark energy responsible for the current acceleration of the Universe. We
parametrise the variation of $\alpha$ with two extra parameters, one defining
the cosmological evolution of the quintessence component and the other fixing
the coupling with the electromagnetic field. The objective of this work is to
constrain these parameters with both astrophysical and local probes. We also
carried out a comparative analysis of how each data probe may constrain our
parametrisation. We performed a Bayesian analysis by comparing the predictions
of the model with observations. The astrophysical datasets are composed of
quasar spectra measurements, including the latest ESPRESSO data point, as well
as Planck observations of the cosmic microwave background. We combined these
with local results from atomic clocks and the MICROSCOPE experiment. The
constraints placed on the quintessence parameter are consistent with a null
variation of the field, and are therefore compatible with a $\Lambda$CDM
cosmology. The constraints on the coupling to the electromagnetic sector are
dominated by the E\"otv\"os parameter local bound. More precise measurements
with ESPRESSO will be extremely important to study the cosmological evolution
of $\alpha$ as it probes an interval of redshift not accessible to other types
of observations. However, for this particular model, current available data
favour a null variation of $\alpha$ resulting mostly from the strong MICROSCOPE
limits.
