Rolling tachyon field models are among the candidates suggested as explanations for the recent acceleration of the Universe. In these models the field is expected to interact with gauge fields and lead to variations of the fine-structure constant α. Here we take advantage of recent observational progress and use a combination of background cosmological observations of Type Ia supernovas and astrophysical and local measurements of α to improve constraints on this class of models. We show that the constraints on α imply that the field dynamics must be extremely slow, leading to a constraint of the present-day dark energy equation of state (1+w₀)<2.4×10⁻⁷ at the 99.7% confidence level. Therefore current and forthcoming standard background cosmology observational probes can't distinguish this class of models from a cosmological constant, while detections of α variations could possibly do so since they would have a characteristic redshift dependence.