There are many possible coordinate systems for describing the Schwarzschild spacetime. Eddington–Finkelstein (EF) coordinates have an especially simple form:
$$ ds^2 = -\left(1 - \frac{r_s}{r}\right) du^2 + 2 du \, dr + r^2 g_\Omega $$
where $g_\Omega = d\theta^2 + \sin^2 \theta d \varphi^2$ is the standard 2-sphere metric in spherical coordinates. Unfortunately, EF coordinates are not regular across the past (white hole) horizon. The same is true for Gullstrand–Painlevé (GP) coordinates.
Kruskal–Szekeres (KS) coordinates have the advantage that they cover the entire manifold of the maximally extended Schwarzschild spacetime, and are well-behaved everywhere outside the physical singularity. The disadvantage is that they are more complex. For instance, their expression needs the Lambert W function.
My question is this: Is there any coordinate system which, like KS, covers the entire maximally extended spacetime but retains a level of simplicity comparable to EF coordinates? I'm especially interested in coordinate systems that are very well-suited for numerical computation and geodesic ray-tracing.