The aeronautical communication has been switched from the congested very high frequency (VHF) band to occupy the L-band, namely, L-band digital aeronautical communication system type I (LDACSI). Critical navigation system in this band is the distance measurement equipment (DME). However, the DME channels provide a severe interference to onboard LDACSI receiver. In this paper, a cognitive radio (CR) network is proposed to allow efficient spectrum reuse with coexisting primary DME channels. In LDACSI-CR network, the proposed spectrum sensing technique is performed by using an adaptive threshold DME energy detector to achieve the best trade-off between the DME signal detection and false alarm. Moreover, dynamic spectrum access, namely, hybrid spectrum access (HSA) is newly introduced to enhance the LDACSI-CR throughput. Joint effect of HSA with different frame structures on the LDACSI throughput is proposed. Besides, optimal power allocation using genetic algorithm that maximizes the LDACSI-CR ergodic throughput for HSA with different frame structures is proposed. In this work, the closed form expression for the probability of DME detection, false alarm, detection error, the LDACSI throughput and data loss are derived. Simulation is performed and the results show the LDACSI throughput and the data loss for the proposed techniques. Besides, ergodic throughput maximization for HSA under constraints of the average LDACSI transmitted power and tolerable interference power on the DME signal is performed. Moreover, simulation results show that the proposed LDACSI-CR system performance provides lower bit error rate and less complexity compared to the previous work.