Low Earth orbit (LEO) satellite communication (SATCOM) networks have gradually been recognized as an efficient solution to enhance ground-based wireless networks. As one of the main characteristics of LEO SATCOM, the beam-edge area could be covered by multiple satellite nodes. In this case, user terminals (UTs) located at the beam-edge have the chance to connect one or more LEO satellites. To develop the diversity gain of multiple nodes in the overlapping area, we propose two high spectral efficiency cooperative transmission strategies, i.e., directly combining (DC) and selection combining (SC). In the DC scheme, signals arrived at the UT simultaneously could be combined into one enhanced signal. For downlink time division multiplexing, the SC scheme enables the UT to select the strongest signal path. Further, as there exists a significant channel gain difference of the beam-center and beam-edge areas, UTs in these two areas can be allocated in one resource block. In this case, we derive co-carriers based on DC and SC, respectively. To deeply analyze the novel methods, we study the ergodic sum-rate and outage probability while the outage diversity gain is further provided. Simulation results show that the co-carrier-based DC method has the ability to provide a higher ergodic sum-rate while the SC method performs better in terms of the outage probability.