Modifying a branched surface to carry a foliation

We consider C1 nonsingular flows on a closed 3-manifold under which there is no transverse disk that flows continuously back into its own interior. We provide an algorithm for modifying any branched surface transverse to such a flow ϕ that terminates in a branched surface carrying a foliation F precisely when F is transverse to ϕ. As a corollary, we find branched surfaces that do not carry foliations but that lift to ones that do.