Melting of the Greenland ice sheet (GrIS) in response to anthropogenic global warming poses a severe threat in terms of global sea level rise of more than 7~m for its complete loss (Allan et al., 2021), but also in terms of North Atlantic freshening that in turn destabilises the Atlantic Overturning Circulation (Boers, 2021). Modelling and paleoclimate evidence suggest that rapidly increasing temperatures in the Arctic can trigger positive feedback mechanisms, and the GrIS is hypothesised to exhibit multiple stable states (Gregory et al., 2020). Consequently, critical transitions are expected when the global mean surface temperature crosses specific thresholds, and there is substantial hysteresis between the alternative stable states (Robinson et al., 2012). Here, we investigate the impact of different overshoot scenarios with varying peak and convergence temperatures for a broad range of warming and subsequent cooling rates. Our results show that both the maximum GMT and the time span of overshooting given global mean temperature (GMT) targets are critical in determining GrIS stability. We find a threshold GMT around 4.5°C above pre-industrial levels for an abrupt loss of the ice sheet, preceded by several intermediate stable states. We show that overshoots of up to 7.5°C GMT above pre-industrial levels might be tolerable if GMTs are subsequently reduced below 1.5°C GMT above pre-industrial levels within a few centuries. However, our results also show that even temporarily overshooting the temperature threshold, without a transition to a new ice sheet state, still leads to dramatic sea-level rise up to several metres during the time span before returning to moderate GMTs.