I'm not sure if this representation is really helpful, but the above sum $S_n$ can be written in terms of the [$q$-digamma function][1] $\psi_q(z)$. Define $q=r\exp(2\pi i/n)$ and $m=\lfloor n/4\rfloor$, with $r<1$ and even $n$, then
\begin{align}
S_n &= 1 + \frac{n}{\pi}\lim_{r\to1^-}
\big[
\psi_q\left( 1 \right)
-\psi_q\left(\tfrac{1}{2}\right)
+\psi_q\left(m+\tfrac{n}{2}+1\right)
-\psi_q\left(m+\tfrac{n}{2}+\tfrac{1}{2}\right)
 \\
&
\qquad{}
-\psi_q(m+1)
+\psi_q\left(m+\tfrac{1}{2}\right)
-\psi_q\left(\tfrac{n}{2}+1\right)
+\psi_q\left(\tfrac{n}{2}+\tfrac{1}{2}\right)
\big].
\label{eq:1}\tag{1}
\end{align}

Mathematica finds this representation if one splits the sum into even and odd parts.

  [1]: https://mathworld.wolfram.com/q-PolygammaFunction.html