The interaction between wind farms based on Doubly-fed induction generators (DFIG) and series capacitors in transmission lines can result in sub-synchronous control interaction (SSCI). Due to the influence of system conditions on the oscillation frequency of SSCI, traditional supplementary sub-synchronous damping control (SSDC) methods, which rely on phase compensation and filtering parameters, become inadequate under varying conditions. To address this issue, this paper proposes a novel supplementary damping-based linear active disturbance rejection control (SDC-LADRC) approach to mitigate SSCI in various scenarios. First, a supplementary damping controller (SDC) is designed to enhance positive damping on the rotor side. Next, the combination of LADRC control and SDC replaces rotor-side converter (RSC) inner loop control. The SDC-LADRC controller leverages the linear extended state observer (LESO) within the LADRC framework to estimate subsynchronous disturbances and provide real-time feedback compensation. This approach effectively suppresses SSCI at full compensation levels, outperforming traditional SSDC methods. Finally, the effectiveness of proposed control method is verified through MATLAB/Simulink simulation tools.