Time Delay Reservoir (TDR) can exhibit effects of high dimensionality and short-term memory based on delay differential equations (DDEs), as well as having hardware-friendly characteristics. However, the predictive performance and memory capacity of the standard TDRs are still limited, and dependent on the hyperparameter of oscillation function. In this paper, we first analyze these limitations and their corresponding reasons. We find that the reasons for such limitations are fused by two aspects, which are the trade-off between the strength of self-feedback and neighboring-feedback caused by neuron separation, as well as the unsuitable order setting of the nonlinear function in DDE. Therefore, we propose a new form of TDR with second-order time delay and a hyperparameter-free nonlinear function to overcome such limitations. Our experiments show that the proposed approach achieves better predictive performance and memory capacity compared with the standard TDR.