The answer is no. For example the group $S_3\times C_3$ occurs (as TransitiveGroup(6,5)) as the Galois group of an (automatically tame) extension of $\mathbb{Q}_5$ (and also for infinitely many other primes), see https://www.lmfdb.org/padicField/5.6.4.2 . The point stabilizer (namely, the diagonal subgroup $C_3$) is contained in a maximal subgroup of index 2, but not one of index 3, meaning that there is a tame degree 6 extension not containing a degree 3 subextension.

The answer is no. For example the group $S_3\times C_3$ occurs (as TransitiveGroup(6,5)) as the Galois group of an (automatically tame) extension of $\mathbb{Q}_5$ (and more generally over $\mathbb{Q} _p$ for all odd primes $p\equiv 2$ mod $3$, by taking the compositum of the unramified degree-$3$ extension with the splitting field of $x^3-p$), see https://www.lmfdb.org/padicField/5.6.4.2 . The point stabilizer (namely, the diagonal subgroup $C_3$) is contained in a maximal subgroup of index 2, but not one of index 3, meaning that there is a tame degree 6 extension not containing a degree 3 subextension.