Let M be an orientable smooth n-manifold and $p: TM \rightarrow M$ be its tangent bundle. In addition, suppose $f: E \rightarrow M$ is a Hurewicz fibration. Write $e(M) \in H^n(M)$ for the Euler class. If the pull-back $f^*(e(M)) \in H^n(E)$ is zero, does that imply that $f$ lifts to a map $g: E \rightarrow TM$ such that $g(x)$ is a nonzero vector for every $x \in E$?
Edit: I'm aware that if $e(M)$ itself is zero, then even the identity lifts, in other words, the tangent bundle has a section. Knowing that $e(M)$ is the only obstruction for this section, the question I'm interested in is: even if $e(M) \neq 0$, if this obstruction class pulls back to zero somewhere, does that mean that there is no obstruction left in that place for a section of the pulled-back tangent bundle?