Avoiding the spherical polar coordinate singularity on $S^2$ by using a double cover?

Question

Is it possible to avoid the spherical polar coordinate singularity on $S^2$ by taking the coordinates as they originally are on $T^2$, i.e. ranging from $0$ to $2\pi$ mod $2\pi$? How would one incorporate this in practice? The position vector to a point on a unit $S^2$ is ${\bf r} = {\bf e}_{\rho}$, where ${\bf e}_{\rho}$ is a unit vector in the direction of increasing radius. With the usual single cover, ${\bf e}_{\theta}$, where $\theta$ indicates the azimuth, becomes ambiguous at the North and South poles. How exactly is this avoided with the double cover?