The answer is yes, given that the domain has a real analytic boundary.
Every biholomorphism $\phi$ of $D^2(r)^o \subset \mathbb{C}$ which has a continuous extension to $D^2(r)$ extends to a biholomorphism of the Riemann sphere. You can use e.g. a Schwarz reflection along $\partial D^2(r)$.
For a more general domain, we can do a similar extension given that the boundary is real analytic.