The Hardening Soil (HS) model has been widely used for the simulation of geotechnical problems. Its popularity has been sustained by its ability to simulate some of the key features of soil response, such as stress dependency, dilatancy and highly non-linear pre-failure response, associated with its relatively simple and clear calibration process based on standard laboratory and field test results and well-established empirical formulas. To improve its ability to deal with unloading, the model was later extended with a small-strain stiffness formulation, leading to the HS-small model. It has become a natural choice for simulations involving retaining walls, shafts and tunnels. Moreover, given the hysteretic nature of the small-strain stiffness formulation, the HS-small model has also emerged as a suitable choice for the simulation of wave propagation and vibration problems within the small to medium strain range. However, as many other constitutive models based on non-linear hysteretic elasticity (or plasticity), it has been recently found that HS-small can suffer from overshooting issues, which are characterised by an overprediction of stiffness after the occurrence of minor unloading-reloading cycles, due to the reset of the strain history tensor. Besides being often found in dynamic problems (e.g. vibrations induced by pile driving, traffic loading or machine operation), these issues may generally affect all boundary-value problem simulations, where numerical oscillations often produce spurious unloading-reloading perturbances. It is therefore imperative to address them. In this paper, the fundamental ingredients of two novel small-strain stiffness formulations addressing overshooting issues in practical applications are presented. The results of laboratory test simulations, including small unloading-reloading cycles interpreted as perturbances, show that both proposed formulations are able to prevent overshooting issues in HS-small. ... Read more