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[over ˜]) in supersymmetric models that exhibit coannihilation between the τ[over ˜] and the lightest neutralino (χ[over ˜]_{1}^{0}), 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 sqrt[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 (χ[over ˜]_{1}^{±}) or next-to-lightest neutralino (χ[over ˜]_{2}^{0}), and the χ[over ˜]_{1}^{0}. The mass of the τ[over ˜] is assumed to be the average of the χ[over ˜]_{1}^{±} and χ[over ˜]_{1}^{0} masses. The data are consistent with standard model background predictions. Upper limits at 95% confidence level are set on the sum of the χ[over ˜]_{1}^{±}, χ[over ˜]_{2}^{0}, and τ[over ˜] production cross sections for Δm(χ[over ˜]_{1}^{±},χ[over ˜]_{1}^{0})=50  GeV, resulting in a lower limit of 290 GeV on the mass of the χ[over ˜]_{1}^{±}, which is the most stringent to date and surpasses the bounds from the LEP experiments.