Cost-friendly differential privacy (CDP) of smart meters can be preserved by an appropriate charging and discharging mechanism that uses rechargeable batteries (RBs) to generate Laplace distributed random noise. However, the existing CDP methods have several issues. Firstly, the maximum discharge rate of an RB requires to vary with the maximal consumption of houses. Secondly, the probability of an RB to charge/discharge depends on the demand, regardless of the state of charge (SoC) of an RB. Thirdly, in extreme SoC (near-empty or almost fully charged) of an RB, no noise added to the demand. To overcome these, we propose a mechanism in which a novel probability density function is designed to generate near Laplace distributed random noise. We also utilize a renewable energy source with small storage in cascade with an RB to enhance performance. Both theoretical analysis and simulations are performed to demonstrate the effectiveness of our proposed method.