Apologies in advance -- I know very little geometry and may have asked a meaningless question/not asked it correctly!
Let $S$ be a compact, connected surface explicitly immersed into Euclidean space $\mathbb R^d$. Let $V\in C^\infty_b(\mathbb R^d\to \mathbb R^d)$ and consider the corresponding ODE $$\dot y^x_t=V(y_t^x),\quad y_0^x =x\in\mathbb R^d.$$
Suppose there exists $\varepsilon, C>0$ such that for all $t\leq\varepsilon$ and $x\in S$, $$d(y_t^x, S)\leq Ct^2,$$ where $d$ denotes the Euclidean metric. (This is what I mean by 'almost tangential'; the solution to the ODE can fall off the surface, but only with a small error.)
Is there some sort of minimal adjustment one can make to the vector field, $V$, so as to ensure the solution to the ODE remains on the surface? All sorts of comments/vague answers appericiated -- many thanks!