Studies of dyadic Green's functions of a planar-layered anisotropic medium started to attract interest in the 1960's. As an important type, the uniaxial one has drawn much attention. Then, investigation on the dyadic Green's function of a stratified uniaxial medium mainly focuses on two different cases: when the optical axis is parallel to the direction of strata, and when the optical axis is perpendicular to this direction. In the first case, one is able to use methods holding for the stratified isotropic medium. However, if the optical axis is perpendicular to the direction of strata, the dispersion relations could be elliptic and no longer the same as those in the isotropic layered medium, thus one has to match boundary conditions within the kx-ky plane instead of within the kρ plane as in the isotropic case. The so-called propagator matrix methods first construct their own state equations, and then use them in order to construct a propagator matrix that relates the total fields on the different boundaries to numerically solve for the different wave modes. Yet the matrix methods proposed in these references suffer from numerical instability when frequencies are high (appreciated on a case by case basis) and/or thicknesses of the layers electrically large. To overcome it, a recurrence relation has been proposed. But to use this relation, the eigenvalues of the state equation must be sorted out by comparing their real parts for every sample in the lateral spectrum plane, which is made by looping on the sampled spectrum, costing much CPU time since the number of the spectral samples is large. In this contribution, new recurrence relations are introduced for the uniaxial planar-layered medium wherein optical axes are perpendicular to the stratified direction. These relations only require to know the signs of the real parts of the eigenvalues. For the planar-layered uniaxial medium, one is able to give the explicit expressions of the four eigenvalues of the state equation, thus one knows the sign of the real part of every eigenvalue at every spectrum point. With such an information, by using the new recurrence relations, there is no more need to compare the real part of the four eigenvalues at every spectral point and thus one saves computational resources (CPU time and memory). Appropriate examples for multi-layered electrically uniaxial composites pertinent to real-world applications will illustrate that the new recurrence relations share the same numerical stability as earlier works.