In this paper, a Resistor Hopping (RH) scheme with the addition of biases is proposed for secure Kirchhoff Law Johnson-Noise (KLJN) communication. The RH approach enables us to increase the bit rate of secure communication between Alice and Bob, while also ensuring that the inherent unconditional security of KLJN is satisfied. The biases are added to the proposed scheme to better distinguish between Gaussian distributed noises in terms of their means, rather than just using variances. Throughout the paper, we strive to minimize biases to achieve a power-efficient scheme. For the detection part of the proposed algorithm, a Maximum-Likelihood (ML) detector is derived. The separability condition of Gaussian distributions is investigated, along with the provision of a threshold-based detector that offers both simple and optimal thresholds in terms of minimizing the error probability. Some analysis of the proposed RH-KLJN communication scheme is provided, including Physical Layer Security (PLS) equations. Simulation results demonstrate the advantages of the proposed scheme over the classical KLJN scheme, offering a higher data rate and lower bit error probability at the expense of increased complexity.