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The X-ray emission of gamma-ray bursts (GRBs) is often characterized by an initial steep decay, followed by a nearly constant emission phase (so called "plateau") which can extend up to thousands of seconds. While the steep decay is usually interpreted as the tail of the prompt gamma-ray flash, the long-lasting plateau is commonly associated to the emission from the external shock sustained by energy injection from a long lasting central engine. A recent study proposed an alternative interpretation, ascribing both the steep decay and the plateau to high-latitude emission (HLE) from a "structured jet" whose energy and bulk Lorentz factor depend on the angular distance from the jet symmetry axis. In this work we expand over this idea and explore more realistic conditions: (a) the finite duration of the prompt emission, (b) the angular dependence of the optical depth and (c) the lightcurve dependence on the observer viewing angle. We find that, when viewed highly off-axis, the structured jet HLE lightcurve is smoothly decaying with no clear distinction between the steep and flat phase, as opposed to the on-axis case. For a realistic choice of physical parameters, the effects of a latitude-dependent Thomson opacity and finite duration of the emission have a marginal effect on the overall lightcurve evolution. We discuss the possible HLE of GW170817, showing that the emission would have faded away long before the first Swift-XRT observations. Finally, we discuss the prospects for the detection of HLE from off-axis GRBs by present and future wide-field X-ray telescopes and X-ray surveys, such as eROSITA and the mission concept THESEUS.