Suppose $s\in E$ is a section of a positive degree vector bundle on an algebraic variety $X$. Then naturally, I can always obtain a short exact sequence $0\to O_X\to E\to F\to 0$ defiend by $s$ and $F$ is only necessarily coherent. Now if I assume $E$ has positive degree, is it possible to obtain a short exact sequence $0\to L\to E\to G\to 0$ for some other line bundle $L$ in a nice way (here I allow $G$ to be a coherent sheaf or something weaker again).
For example, I would hope that if $s\in \Gamma(X,E)$ is a global section, then $s$ would define an inclusion $L\to E$ from some line bundle $L$ which is not trivial.
Here, I've left some freedom to deal with cases: (1) $X$ is a curve so there is hope to make $F$ itself a vector bundle (I do not care for this case as much), (2) $X$ is a surface so that $L$ should be the line bundle associated to some some curves, and (3) $X$ is higher dimensional which I am uninterested in and expact there to be difficulties.
Already for surfaces it is not always possible to express a vector bundle as an extension of vector bundles of smaller rank. For instance, it can be that the rank of $E$ is 2, but $c_2(E)$ does not belong to the image of the intersection map $\operatorname{Pic}(X) \times \operatorname{Pic}(X) \to \operatorname{CH}^2(X)$. Then $E$ is not an extension of line bundles.