A branched surface that meets the torus boundary of a compact 3-manifold transversely (in a train track \tau) can sometimes be ``upgraded'' to a branched surface that fully carries CTFs that strongly realize all boundary slopes except one. We give a condition on \tau that guarantees that such an upgrade is possible. This approach succeeds for all alternating and Montesinos knots without L-space surgeries, for certain Murasugi sums, and for all nontrivial connected sums of alternating knots, Montesinos knots, or fibered knots. This is joint work with Charles Delman.