The α-attractor inflationary models are nowadays favored by CMB Planck observations. Their similarity with canonical quintessence models motivates the exploration of a common framework that explains both inflation and dark energy. We study the expected constraints that next-generation cosmological experiments will be able to impose for the dark energy α-attractor model. We systematically account for the constraining power of SNIa from WFIRST, BAO from DESI and WFIRST, galaxy clustering and shear from LSST and Stage-4 CMB experiments. We assume a tensor-to-scalar ratio, 10-3 < r < 10-2, which permits to explore the wide regime sufficiently close, but distinct, to a cosmological constant, without need of fine tunning the initial value of the field. We find that the combination S4CMB + LSST + SNIa will achieve the best results, improving the FoM by almost an order of magnitude; respect to the S4CMB + BAO + SNIa case. We find this is also true for the FoM of the w0-wa parameters. Therefore, future surveys will be uniquely able to probe models connecting early and late cosmic acceleration.