A thermal-assisted electroforming (TAE) method is proposed to address the current overshoot issue and improve the resistive switching (RS) performance of electrochemical metallization (ECM) memory with a Cu/amorphous carbon (a-C)/Pt structure. In the initial electroforming process, thermal treatment can promote the electrochemical formation of metallic conductive filament (CF). The required electroforming voltage is reduced, and the undesirable overshoot current is suppressed. As a result, the RS performance of the cell is improved, including reduced RS parameter fluctuations, enlarged off/on ratio, and enhanced cycling endurance. The cell can be operated with a low compliance current (ICC) of 50 μA, which is among the lowest value of reported a-C-based ECM memories. Benefit from the low ICC, multilevel memory with five nonvolatile resistance states in a single cell is obtained. The main role of the TAE method is interpreted as to avoid the extreme high-temperature caused by the overshoot current. Conductive-atomic force microscopy mapping implies that the suppression of overshoot current can avoid over-injection of Cu cations into the a-C layer, facilitating CF with a simple structure and low randomness. The present work offers a feasible approach for addressing the current overshoot issue and improving the performance of ECM memory.