A measurement of the top-quark pole mass $ {m}_t^{\textrm{pole}} $ is presented in $ t\overline{t} $ events with an additional jet, $ t\overline{t} $ + 1-jet, produced in pp collisions at $ \sqrt{s}=13 $ TeV. The data sample, recorded with the ATLAS experiment during Run 2 of the LHC, corresponds to an integrated luminosity of 140 fb$^{−1}$. Events with one electron and one muon of opposite electric charge in the final state are selected to measure the $ t\overline{t} $ + 1-jet differential cross-section as a function of the inverse of the invariant mass of the $ t\overline{t} $ + 1-jet system. Iterative Bayesian Unfolding is used to correct the data to enable comparison with fixed-order calculations at next-to-leading-order accuracy in the strong coupling. The process $ pp\to t\overline{t}j\left(2\to 3\right) $, where top quarks are taken as stable particles, and the process $ pp\to b\overline{b}{l}^{+}\nu {l}^{-}\overline{\nu}j\left(2\to 7\right) $, which includes top-quark decays to the dilepton final state and off-shell effects, are considered. The top-quark mass is extracted using a χ$^{2}$ fit of the unfolded normalized differential cross-section distribution. The results obtained with the 2 → 3 and 2 → 7 calculations are compatible within theoretical uncertainties, providing an important consistency check. The more precise determination is obtained for the 2 → 3 measurement: $ {m}_t^{\textrm{pole}}=170.7\pm 0.3\left(\textrm{stat}.\right)\pm 1.4\left(\textrm{syst}.\right)\pm 0.3\left(\textrm{scale}\right)\pm 0.2\left(\textrm{PDF}\oplus {\alpha}_{\textrm{S}}\right) $ GeV, which is in good agreement with other top-quark mass results.[graphic not available: see fulltext]