Let $M$ be a smooth manifold with a smooth submanifold $N$ and vector field $v$ that's never parallel to $N$. Suppose we want to extend some function $f: N \to \mathbb{R}$ to a function $F$ on all of $M$, subject to the constraint that $dF(v)>-1$ everywhere. If $\gamma:[t_1,t_2]\to M$ is an integral curve of $v$ beginning and ending on $N$, then we must have $$f(\gamma(t_2))-f(\gamma(t_1))=\int_{t_1}^{t_2} dF(\dot \gamma(t)) \, dt>\int_{t_1}^{t_2}\dot (-1)\, dt=t_1-t_2.$$ Is this the only form of obstruction to finding an extension $F$?
Note: I'm particularly interested in the case where $M$ is a closed 3-manifold with contact structure $\ker \alpha$, $N$ is a knot, and $v=R_\alpha$ is the (nonvanishing) Reeb vector field associated to $\alpha$.