The electrostatic problem for a multilayered particle having spherical or non-confocal spheroidal layer boundaries arranged in an arbitrary order are solved. Scalar potentials in the vicinity of the layer boundaries are expanded using spherical or spheroidal bases, respectively, and the potential expansions in each layer are matched using expansions of spheroidal functions in terms of spherical functions and vice versa. Using the extended boundary condition method with a spheroidal basis (two schemes of this method are employed), infinite systems of linear algebraic equations for the expansion coefficients of the unknown fields are obtained. The solution to the problem is formulated in terms of a T-matrix, which is convenient for considering particle ensembles. It is shown that the resulting T-matrix must be symmetric, which is important for controlling the accuracy of computations. In this work, the Rayleigh approximation, which is well-known for confocal spheroids, is extended to a significantly wider class of particles, taking into account that the T-matrix element T₁₁ is directly proportional to the particle's polarizability. In the particular case of two-layer spheroidal particles with a spherical core, considered previously, the performed calculations demonstrate the high efficiency of the proposed solution to the electrostatic problem. Keywords: Rayleigh approximation, electrostatics, layered spheroids, T-matrix, particle polarizability.