Precisely and automatically detecting the cough sound is of vital clinical importance. Nevertheless, due to privacy protection considerations, transmitting the raw audio data to the cloud is not permitted, and therefore there is a great demand for an efficient, accurate, and low-cost solution at the edge device. To address this challenge, we propose a semi-custom software-hardware co-design methodology to help build the cough detection system. Specifically, we first design a scalable and compact convolutional neural network (CNN) structure that generates many network instances. Second, we develop a dedicated hardware accelerator to perform the inference computation efficiently, and then we find the optimal network instance by applying network design space exploration. Finally, we compile the optimal network and let it run on the hardware accelerator. The experimental results demonstrate that our model achieves 88.8% classification accuracy, 91.2% sensitivity, 86.5% specificity, and 86.5% precision, while the computation complexity is only 1.09M multiply-accumulation (MAC). Additionally, when implemented on a lightweight field programmable gate array (FPGA), the complete cough detection system only occupies 7.9K lookup tables (LUTs), 12.9K flip-flops (FFs), and 41 digital signal processing (DSP) slices, providing 8.3 GOP/s actual inference throughput and total power dissipation of 0.93 W. This framework meets the needs of partial application and can be easily extended or integrated into other healthcare applications.