The first search for supersymmetry in events with an experimental signature of one soft, hadronically decaying τ 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 τ leptons (τ) in supersymmetric models that exhibit coannihilation between the τ and the lightest neutralino (χ10), 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 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 (Δm) between the chargino (χ1±) or next-to-lightest neutralino (χ20), and the χ10. The mass of the τ is assumed to be the average of the χ1± and χ10 masses. The data are consistent with standard model background predictions. Upper limits at 95% confidence level are set on the sum of the χ1±, χ20, and τ production cross sections for Δm(χ1±,χ10)=50 GeV, resulting in a lower limit of 290 GeV on the mass of the χ1±, which is the most stringent to date and surpasses the bounds from the LEP experiments.