Suppose you have a vector bundle $E \to B$ with projection $p$. Consider a local trivialization $h_\alpha: U_\alpha \times \mathbb{R} \to p^{-1}(U_\alpha)$ and take the map $i_\alpha: U_\alpha \to U_\alpha \times \mathbb{R}$ given by $x \rightsquigarrow (x,0)$. The $U_\alpha$ cover $B$.
Is it true that we can always arrange the local trivializations so that for any $\alpha$, $p \circ h_\alpha \circ i_\alpha=Id_{U_\alpha}$?