The posted salary range was nearly double at the top end and the recruiter wouldn't narrow it

Range on the posting spanned almost 2x from bottom to top. On the screen I asked, politely, roughly where someone with my experience would sit in that band. Got "it really depends on overall fit." Asked once more later in the process, same answer in different words.

So I named a number in the top third. They agreed same day, no counter, no flinch. Which almost certainly means I left something on the table.

What actually works to get them to go first when the posted range is that wide? "What's budgeted for this specific role" is the best I've come up with and it worked exactly once.