In high-speed and long-distance networks, TCP NewReno, the most popular version of Transmission Control Protocol (TCP), cannot achieve sufficient throughput owing to the inherent nature of the congestion control mechanism of TCP. Therefore, in order to overcome this limitation, Compound TCP was proposed. Compound TCP can achieve a considerably higher throughput than TCP NewReno in high-speed and long-distance networks. The congestion control mechanism of Compound TCP consists of loss-based and delay-based congestion controls. However, in wireless LAN, the media access control used causes unfairness in the throughput among TCP connections. Compound TCP has the same type of congestion control as TCP NewReno; hence, it is expected that the problem will occur among Compound TCP connections. In this study, we evaluate the performance of Compound TCP for wireless LAN, and demonstrate that the throughput among Compound TCP connections becomes unfair. Then, we propose Compound TCP+, which implements a finer congestion control by detecting a state of slight congestion. Using simulation, we show that in wireless LAN, Compound TCP+ connections achieve fairness and share the bandwidth equally. We also demonstrate through simulation that Compound TCP+ achieves high throughput in a high-speed wired network.