The first search for supersymmetry in events with an experimental signature of one soft, hadronically decaying tau lepton, one energetic jet from initial-state radiation, and large transverse momentum imbalance is presented. These event signatures are consistent with direct or indirect production of scalar tau leptons ((tau) over tilde) in supersymmetric models that exhibit coannihilation between the (tau) over tilde and the lightest neutralino ((chi) over tilde (0)(1)), and that could generate the observed relic density of dark matter. The data correspond to an integrated luminosity of 77.2 fb(-1) of proton-proton collisions at root s = 13 TeV collected with the CMS detector at the LHC in 2016 and 2017. The results are interpreted in a supersymmetric scenario with a small mass difference (Delta m) between the chargino ((chi) over tilde (+/-)(1)) or next-to-lightest neutralino ((chi) over tilde (0)(2)), and the (chi) over tilde (0)(1). The mass of the (tau) over tilde is assumed to be the average of the (chi) over tilde (1)(+/-) and (chi) over tilde (0)(1) masses. The data are consistent with standard model background predictions. Upper limits at 95% confidence level are set on the sum of the (chi) over tilde (+/-)(1). (chi) over tilde (0)(2), and (tau) over tilde production cross sections for Delta m ((chi) over tilde (+/-)(1),(chi) over tilde (0)(1)) = 50 GeV, resulting in a lower limit of 290 GeVon the mass of the (chi) over tilde (+/-)(1), which is the most stringent to date and surpasses the bounds from the LEP experiments.