Yes, $\phi$ is smooth. Indeed, fix any $x\in\partial M$ and take any nieghbourhood $x\in U\subseteq\partial M$ with compact closure. Calling $\nu:\partial M\to TM$ the (unit) inward-pointing normal vector, we can find some $\epsilon>0$ so small that
- the map $\alpha:U\times [0,\epsilon]\to M$, $\alpha(y,t):=\exp(t\nu(y))$ is well-defined and is a diffeomorphism onto its image;
- $\text{dist}(\alpha(y,t),\partial M)=t$ for all $(y,t)\in U\times [0,\epsilon]$.
For any $y\in U$, the curve $t\mapsto\phi\circ\alpha(y,t)$ is a unit-speed geodesic: this is true on the interval $(0,\epsilon]$ by interior smoothness of $\phi$, so it is true on $[0,\epsilon]$ by continuity. Since $\text{dist}(\phi\circ\alpha(y,t),\partial N)=t$, we deduce that it is a minimizing geodesic from $\partial N$ to $\phi\circ\alpha(y,\epsilon)$, implying that $\frac{d}{dt}(\phi\circ\alpha)|_{t=0}\perp\partial M$, i.e. $$ \phi\circ\alpha(y,t)=\exp(t\nu(\phi(y)))\qquad (*)$$ (now $\nu$ denotes the inward-pointing normal in $N$).
Up to shrinking $\epsilon$, we can assume that $(z,t)\mapsto\exp(t\nu(z))$ gives a diffeomorphism from $\phi(U)\times [0,\epsilon]$ onto its image, as well. We call $(\beta,\tau)$ its smooth inverse.
Finally, $\phi|_U$ is smooth (as $\phi(y)=\beta\circ\phi\circ\alpha(y,\epsilon)$ and as $\phi$ is smooth on $\text{int}(M)$), so also $\phi|_{\alpha(U\times[0,\epsilon])}$ (by $(*)$).