In this paper, to meet the demand for the rapid growth of data traffic, we design a high data rate chaotic communication scheme. Consequently, we propose a differential chaos shift keying (DCSK) system with hybrid carrier-code and reference index modulation, which makes full use of time-frequency resources (HCCRIM-TF-DCSK). In this system, orthogonal reference signals are generated using Gram-Schmidt algorithm, and reference index modulation is performed in each time slot of each subcarrier by discarding a chosen reference signal. Additionally, the selected Walsh code in active subcarriers is used to perform the Kronecker product with the reference signal to achieve higher data rates. Furthermore, at the receiver, an efficient joint carrier-code index detection algorithm is proposed, and noise reduction processing is employed to improve the system's bit error rate (BER) performance. The theoretical BER expressions of the HCCRIM-TF-DCSK system over additive white Gaussian noise (AWGN) and multipath Rayleigh fading channels are derived. Furthermore, this paper provides an analysis of the system's data rate, spectral efficiency, energy efficiency and complexity compared to those of advanced competitors. Simulation results confirm the accuracy of the theoretical derivations and the superiority of the system. Overall, the HCCRIM-TF-DCSK system is expected to become a cost-effective wireless communication solution.