The most popular approach in extreme value statistics is the modelling of threshold exceedances using the asymptotically motivated generalised Pareto distribution. This approach involves the selection of a high threshold above which the model fits the data well. Sometimes, few observations of a measurement process might be recorded in applications and so selecting a high quantile of the sample as the threshold leads to almost no exceedances. In this paper we propose extensions of the generalised Pareto distribution that incorporate an additional shape parameter while keeping the tail behaviour unaffected. The inclusion of this parameter offers additional structure for the main body of the distribution, improves the stability of the modified scale, tail index and return level estimates to threshold choice and allows a lower threshold to be selected. We illustrate the benefits of the proposed models with a simulation study and two case studies.