If you map the nth roots of unity $z$ with the function $-W(-z/e)$ you get decent starting points for some root finding algorithm to the roots of the scaled truncated taylor series of $\exp$. Here W is the lambertW function, $e$ is $\exp(1)$ and 'scaled' in 'scaled truncated taylor series of exp' means the following: say if $$s_5(x) = 1+x+x^2/2+x^3/6+x^4/24+x^5/120$$ is the 'truncated taylor series of exp' of degree 5 then we will look at $s_5(5x)$ so we are looking at $s_n(nx)$ in general.
Here is a plot 
for the case $n=33$ (it only works for uneven $n$).
Using the lambert W function comes from formula (1.1) from paper 221 available from here. This formula is:
$$e^{-nz}s_n(nz)=1-\frac{\sqrt{n}}{\tau_n\sqrt{2\pi}}\int_0^z(\zeta e^{1-\zeta})^n\textrm{d}\zeta,~~z\in \mathbb{C}$$
$-W(-z/e)$ is the inverse of $ze^{1-z}$.
How to get a better map from the roots of unity to the roots of this polynomial? Alternatively, is there some infinite sum representation for the roots? There isn't much difference: "applying" LambertW to some start values is pretty much the same as an infinite series.
Here is an octave script for such a plot as the one above (To use lambertw(), as in the script, install the 'specfun' package for octave - or use a more number/function theory centric system than octave).