To complete the enumeration of numerical methods, if the known roots really are good approximations for the roots of the next polynomial, then you can use Weierstraß' method, commonly known as Durand-Kerner method. It updates all roots at once, and is as fast as Newton's method, in fact is is Newton's method for a multidimensional problem.
Aberth-Ehrlich method is related, but it is not certain that the higher computational effort gives an overall faster convergence.
Additionally, since you are doing some kind of ray tracing, the univariate polynomials resulting from the intersection of a polynomial surface with a ray, you can divide the image into 6x6 or 8x8 blocks, compute the roots at a central pixel and then compute the gradients of the surface at the intersection points. You may get inspiration on how to do that by playing with the derivative code generator tapenade.
Using the point $x^*$ and the gradient $\nabla f(x^*)$ for the tangent plane equation $\langle \nabla f(x^*),\,x\rangle=\langle \nabla f(x^*),\,x^*\rangle$, you get better initial approximations by intersecting the rays $x_v(t)=\vec c+t\vec v$ of the other pixels first with the tangent plane.
Check for $\langle \nabla f(x^*),\,v\rangle\approx 0$, this occurs close to the visual contours. These are the critical cases in any approach since they have multiple or very close roots.