The work of Jaco-Shalen and Johansson actually handles manifolds with boundary. The Theorem they prove (in the Language of Johannson's Book) is: 

An irreducible, boundary-irreducible 3-manifold with useful boundary pattern possesses a characteristic submanifold, unique up to isotopy.

The complement of the characteristic submanifold is atoroidal and acylindrical, hence (by Thurston) possesses a unique hyperbolic metric with totallygeodesic boundary.

The characteristic submanifold has the property that every admissibly embedded I-bundle or Seifert fibered space is Isotopic into the characteristic submanifold.

All the notions, including useful boundary Pattern and admissible embedding, are explained in Springer Lecture Notes in Mathematics 761 (Johannson: Homotopy equivalences of 3-manifolds with Boundaries), the above-cited Theorem is Proposition 9.4 in that Book.