Negative reactivity feedback is one of the most important inherent safety characteristics of reactors for suppressing changes in thermal power and component temperatures during accidents. The floating absorber for safety at transient (FAST) has been suggested to improve the coolant temperature coefficient (CTC) in a sodium-cooled fast reactor (SFR) for a better self-stabilization of the reactor system. Previous studies on the FAST application showed that the FAST causes power excursion and temperature oscillations under anticipated transient without scram (ATWS) conditions. This study explains the variables affecting the FAST behavior and oscillations with mathematical expressions. In addition, a damping FAST is proposed to passively remove oscillations without any mechanical system. The damping FAST increases the damping ratio by reducing the pin diameter at a height where the difference between the FAST bottom elevation and the equilibrium position is maximum. The performance evaluation for the damping FAST is conducted in an innovative sodium-cooled fast reactor (iSFR) under ATWS conditions. The 7-mm damping FAST unilaterally inserts the negative reactivity in unprotected transient overpower event, dramatically lowering the core power without oscillations. While the previous FAST shows oscillatory behavior repeatedly approaching the coolant boiling temperature, the damping FAST stably keeps the coolant temperature below 800 °C. Moreover, the damping FAST lowers the maximum coolant temperatures for other ATWS events. These findings show the potential of the FAST system to significantly improve the inherent safety of SFRs.