We report a measurement of a beam–target double-polarisation observable (E) for the γ→n→(p)→K+Σ−(p) reaction. The data were obtained impinging the circularly-polarised energy-tagged photon beam of Hall B at Jefferson Lab on a longitudinally-polarised frozen-spin hydrogen deuteride (HD) nuclear target. The E observable for an effective neutron target was determined for centre-of-mass energies 1.70≤W≤2.30 GeV, with reaction products detected over a wide angular acceptance by the CLAS spectrometer. These new double-polarisation data give unique constraints on the strange decays of excited neutron states. Inclusion of the new data within the Bonn-Gatchina theoretical model results in significant changes for the extracted photocouplings of a number of established nucleon resonances. Possible improvements in the PWA description of the experimental data with additional "missing" resonance states, including the N(2120)3/2− resonance, are also quantified.