Contour integral $\oint \frac{\mathrm{d}z}{2 \pi i z} \log(w+z)$

Question

Let $\log:\mathbb{C} \to \mathbb{R} \times i (-\pi,\pi]$ denote the principal branch of the complex logarithm. For $\omega \in \mathbb{C}$, I would like to calculate the contour integral \begin{equation} \oint_{\lvert z\rvert = 1} \frac{\mathrm{d}z}{2 \pi i z} \log(w+z), \end{equation} I have been able to do this for all $\omega$ in $\mathbb{C}$ with $|\omega|>1$. Indeed, for such $\omega$ the function $\log(w+z)$ is holomorphic for $z \in \mathbb{D}$, and hence the Cauchy integral formula tells us that \begin{equation} \oint_{S^1} \frac{\mathrm{d}z}{2 \pi i z} \log(w+z) = \log(w) \qquad \text{for $|\omega| > 1$}. \end{equation} What about when $|\omega| \leq 1$?