One of the unique traffic features of low earth orbit (LEO) satellite networks is nonuniform traffic load distribution. Moreover, the traffic distribution is time-variant since each individual satellite moves along its orbit plane and the earth rotates on its axis simultaneously. A call in a LEO satellite network has to be rerouted periodically as a result of the satellite handover. In this paper, we propose an adaptive routing protocol in LEO satellite networks to provide better quality of service regardless the traffic distribution in the network. The proposing routing protocol distributes the traffic load of congested areas, while it optimizes the suboptimally routed path without the overhead of end-to-end rerouting. In addition, it reduces the handover blocking probability using a different threshold value for the call admission of handover calls.