Let $p$ be a prime number and $\mathbb{Q}_p$ be the $p-$adic rational field. Let $E/\mathbb{Q}_p$ be a fixed finite extension. On this site, I define a finite extension $F/E$ is "good" if there exists a intermediate extension $K/E$ of $F/E$ such that $K/E$ is totally ramified and $F/K$ is unramified.

My question is, for any finite extension $L/E$, does there exist a "good" extension $F/E$, such that $L$ is contained in $F$? Since we only care about the inclusion $L\subset F$, we can enlarge $L$, such that $L/E$ is a Galois extension.

If $L/E$ is an abelian extension, the answer to my question is yes. From class field theory, we can construct the Lubin-Tate extension $E_\pi/E$, which is totally ramified. The maximal abelian extension of $E$ is obtained by the composite of $E_\pi$ and the maximal unramified extension of $E$. Hence we can find a "good" extension $F/E$ such that $F\supset L$.

Since this question looks rather fundamental, I wonder if this question was considered and solved by anyone else before. Any help will be appreciated.