Vector field on a geodesic

Question

Let $M$ be a complete Riemannian manifold and $f:M\rightarrow\mathbb{R}$ be a nonconstant smooth function. If $\gamma:[0,1]\rightarrow M$ be a geodesic without any conjugate point on $M$ and it does not pass through any critical point of $f$, then is it possible that there exists a smooth vector field $W(t)$ on $\gamma(t)$ such that $df_{\gamma(t))}(W(t))>0$?
Thank you.