Words that give rise to an enumeration of elements of the symmetric group

Let $$\mathbb{S}_m$$ be the symmetric group on $$m$$ letters. Let $$n=m-1$$. Let $$\mathbf{w}=a_1\cdots a_r$$ be a word on the alphabet $$\{1,\ldots,n\}$$. We say that $$\mathbf{w}$$ gives rise to an enumeration of elements of the symmetric group $$\mathbb{S}_m$$ if $$1,s_{a_r},s_{a_{r-1}}s_{a_r},\ldots,s_{a_2}\cdots s_{a_r},s_{a_1}\cdots s_{a_r}$$ are all distinct elements of the symmetric group $$\mathbb{S}_m$$.

For example, $$32323132323132323$$ gives rise to an enumeration of elements of the symmetric group $$\mathbb{S}_4$$, and I think it is (one of) the longest possible such words.

Q. What is the length $$r(n)$$ of the longest word which gives rise to an enumeration of elements of the symmetric group $$\mathbb{S}_m$$. Is it always possible to find such a word of length $$m!$$, or only for $$\mathbb{S}_3$$?