Suppose $E$ is an oriented vector bundle over a closed manifold $M$ where the rank of $E$ equals to $n:=\dim M$ and $n$ is odd. Classical obstruction theory tell us, that $E$ admits a nowhere vanishing section, say $\sigma$, which is equivalent to a cross-section in the sphere bundle $\Sigma E$.
Now, suppose $\sigma'$ is another cross-section of $\Sigma E$. Is there a (obstruction theoretic) method to tell if $\sigma$ and $\sigma'$ are homotopic as sections?