Here is an answer of sorts; it is not completely canonical though. First of all, you have to pick a conformal or hyperbolic structure on the fiber $\Sigma$. This can be made almost canonical, since there is a unique $\phi$-invariant (real) Teichmuller geodesic. In any case, you get $\Sigma={\mathbb H}^2/\Gamma$, where $\Gamma$ is $\pi_1(\Sigma)$ acting on the hyperbolic plane (isometrically, freely, properly). The map $\phi$ induces a canonical $\Gamma$-equivariant homeomorphism $f: S^1\to S^1$ (of the boundary circle of the hyperbolic plane). Now, use the Douady-Earle extension to extend $f$ to a  $\Gamma$-equivariant diffeomorphism $F$ of the hyperbolic plane. Lastly, project $F$ to a diffeomorphism $\phi$ of the surface $\Sigma$.