We report a measurement of a beam-target double-polarisation observable (E) for the → 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