Let screenreader know about content prior to focus location

Consider the Amazon search bar: it has a drop-down at the beginning that lets you select which department to search in, then the search input, and then the search submit button.

I'd like to have a similar search interface, but have it display in a modal that opens up when the user clicks a search button. For most people - interacting visually with the site - throwing focus on the input when search opens will be the best UX. The subset who want to select a result category other than "all" can click or shift-tab to it.

However, we can't use that approach if it means screenreader users won't know the select even exists. Is there a way to let the screenreader user know the select drop-down is there when the focus is set on the input, later in the HTML?

The closest I have come to a solution is to use an aria-describedby attribute on the input, pointing at visually-hidden text describing the select and its relationship to the input, but my attempts at that text are clunky and long and not very clear.