The first search for supersymmetry in events with an experimental signature of one soft, hadronically decaying $\ensuremath{\tau}$ 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 $\ensuremath{\tau}$ leptons ($\stackrel{\texttildelow{}}{\ensuremath{\tau}}$) in supersymmetric models that exhibit coannihilation between the $\stackrel{\texttildelow{}}{\ensuremath{\tau}}$ and the lightest neutralino (${\stackrel{\texttildelow{}}{\ensuremath{\chi}}}_{1}^{0}$), and that could generate the observed relic density of dark matter. The data correspond to an integrated luminosity of $77.2\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of proton-proton collisions at $\sqrt{s}=13\text{ }\text{ }\mathrm{TeV}$ 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 ($\mathrm{\ensuremath{\Delta}}m$) between the chargino (${\stackrel{\texttildelow{}}{\ensuremath{\chi}}}_{1}^{\ifmmode\pm\else\textpm\fi{}}$) or next-to-lightest neutralino (${\stackrel{\texttildelow{}}{\ensuremath{\chi}}}_{2}^{0}$), and the ${\stackrel{\texttildelow{}}{\ensuremath{\chi}}}_{1}^{0}$. The mass of the $\stackrel{\texttildelow{}}{\ensuremath{\tau}}$ is assumed to be the average of the ${\stackrel{\texttildelow{}}{\ensuremath{\chi}}}_{1}^{\ifmmode\pm\else\textpm\fi{}}$ and ${\stackrel{\texttildelow{}}{\ensuremath{\chi}}}_{1}^{0}$ masses. The data are consistent with standard model background predictions. Upper limits at 95% confidence level are set on the sum of the ${\stackrel{\texttildelow{}}{\ensuremath{\chi}}}_{1}^{\ifmmode\pm\else\textpm\fi{}}$, ${\stackrel{\texttildelow{}}{\ensuremath{\chi}}}_{2}^{0}$, and $\stackrel{\texttildelow{}}{\ensuremath{\tau}}$ production cross sections for $\mathrm{\ensuremath{\Delta}}m({\stackrel{\texttildelow{}}{\ensuremath{\chi}}}_{1}^{\ifmmode\pm\else\textpm\fi{}},{\stackrel{\texttildelow{}}{\ensuremath{\chi}}}_{1}^{0})=50\text{ }\text{ }\mathrm{GeV}$, resulting in a lower limit of 290 GeV on the mass of the ${\stackrel{\texttildelow{}}{\ensuremath{\chi}}}_{1}^{\ifmmode\pm\else\textpm\fi{}}$, which is the most stringent to date and surpasses the bounds from the LEP experiments.