I do not know how well this compares to other methods, but it is one I have devised for the purpose of computing the polylogarithm for values of z near the unit circle. In the case of the dilogarithm, the group of anharmonic ratios allows one to reduce the computation in the general case to a fundamental region for that group. However that still appears to involve computation for values near (or on) the unit circle, in particular the cube roots of -1 in the right half plane are fixed points for that group, so one needs to compute those. And nearby values.
A rational approximation to the polylogarithm can be had with poles in the branch cut (where they ought to be for any decent approximation). Runge's theorem guarantees there will be many.
We arrived at this one:
$$\text{Li}_{s}(z)\approx \frac{\pi z}{Γ(s)\sqrt{N}}
\sum_{k=-N}^{N}\frac{e^{k\pi/\sqrt{N}}}{(e^{k\pi/\sqrt{N}}+1-z)(1+e^{k\pi/\sqrt{N}})}
\log^{s-1}(1+e^{k\pi/\sqrt{N}})$$
$$= z\sum_{k=-N}^{N}\frac{w_{k}}{λ_{k}-z}$$
which is a Riemann sum for the integral representation
$$\text{Li}_{s}(z)=\frac{z}{Γ(s)}\int_{-\infty}^{\infty} \frac{e^{\tau}}{(e^{\tau}+1-z)(1+e^{\tau})} \log^{s-1}(1+e^{\tau})d\tau$$
which follows from a simple change of variables in the well known integral representation
$$\text{Li}_{s}(z)=\frac{1}{Γ(s)}\int_0^\infty \frac{ze^{-t}}{1-ze^{-t}}t^{s-1}dt$$
which can be obtained easily from the Laplace transform.