The High Energy Stereoscopic System (H.E.S.S.) is one of the currently operating Imaging Atmospheric Cherenkov Telescopes. H.E.S.S. operates in the broad energy range from a few tens of GeV to more than 50 TeV reaching its best sensitivity around 1 TeV. In this contribution, we present an analysis technique, which is optimised for the detection at the highest energies accessible to H.E.S.S. and aimed to improve the sensitivity above 10 TeV. It includes the employment of improved event direction reconstruction and gamma-hadron separation. For the first time, also extensive air showers with event offsets up to 4.5 deg from the camera centre are considered in the analysis, thereby increasing the effective Field-of-View of H.E.S.S. from 5 deg to 9 deg. Key performance parameters of the new high-energy analysis are presented and its applicability demonstrated for representative hard-spectrum sources in the Milky Way.