The spectrograph ESPRESSO obtained updated limits on the variation of the fine-structure constant, $\alpha$, through measurements along the line of sight of a bright quasar with unprecedented precision and accuracy. These impose 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 make a comparative analysis on how each data probe may constrain our parametrisation. We perform a Bayesian analysis by comparing the predictions of the model with observations. The astrophysical datasets are composed of quasars spectra measurements, including the latest ESPRESSO data point, as well as Planck observations of the cosmic microwave background. We combine them 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, i.e. compatible with a $\Lambda$CDM cosmology. The constraints on the coupling to the electromagnetic sector are dominated by the Eötvös parameter local bound. The ESPRESSO data point is extremely important to study the cosmological evolution of $\alpha$ as it probes an interval of redshift not accessible to other observations. However, for this particular model, current available data favours a null variation of $\alpha$ resulting mostly from the strong MICROSCOPE limits.