Scattering of an on- or off-axis focused Gaussian beam by a coated spheroidal particle (either a prolate or an oblate) is presented. A technique combines the plane-waves spectrum and the T-matrix methods are used to solve the scattering problem. Results are shown for different beam focusing positions with respect to the particle and for different particle's shapes (sphere, and prolate and oblate spheroids) of large size parameters, and for different core's radius or shell's thickness. The angular scattering intensities are affected by the changes of the core's radius in the case of spheroidal particle more than that for the case of a spherical particle resonates at a low-order mode. As the beam moves farther away from the center of the spheroid the effect of the core on the scattering intensity decreases. A coated spheroidal particle is much affected by changing the beam focusing position than that in the case of a homogeneous spheroidal particle.