The answer is yes.
Given that the domain has a real analytic boundary the construction is standard. A Schwarz reflection can be performed locally near each boundary point, which together with normal analytic continuation produces the sought extension.
My first claim is that we can cover the domain $U$ by closed analytic discs contained entirely in $U$. Here it is crucial that the boundary is smooth, in order to produce analytic discs which are tangent to the boundary at every point. (In order to construct such discs, one can e.g. use a smooth approximation of a closed curve by analytic such curves.) Since $\phi$ restricted to each such disc has an extension by the above procedure, standard analytic continuation thus produces the sought extension of $\phi$.
(As a particular case, which however is not so relevant given the new general formulation of the question: 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. Here we use a Schwarz reflection along $\partial D^2(r)$.)