The effect of the azimuthal angle φ of the wave vector k on the propagation characteristics of the superluminous L-O mode waves (together with a case of the R-X mode) during different geomagnetic activities using a three-dimensional (3D) ray-tracing method is investigated. This work is primarily an extension of our previous two-dimensional study in which the wave azimuthal angle was not considered. We present numerical simulations for this mode which is generated in the source cavity along a 70° night geomagnetic field line at the specific altitude of 1.5RE (where RE is the Earth's radius). It is found that, as in the two-dimensional case, the trajectory of L-O mode starting in the source meridian plane (or the wave azimuthal angle φ = 180°) can reach the lowest latitude; whereas it basically stays at relatively higher latitudes starting off the source meridian plane (or φ=180°). The results reveal that under appropriate conditions, the superluminous L-O mode waves may exist in the radiation belts of the Earth, but this remains to be supplemented by observational data.