Obliquely. The answer to your question about how to best establish that your story is set somewhere in the near future without being explicit, is to obliquely tie it to a well known event in the past.
Without knowing more about your story, its hard to make more precise suggestions by the idea is to have something in the story that a character can comment on or react to
"It has been more than thirty years since the twin towers fell, why are people ... <insert something that reveals character qualities, advances plot, or builds on what is at stake in the story."
or "the pandemic was over like ten years ago, or there about, ... <insert same idea about character, place, setting, stakes, tension, whatever you want>
This is a general worldbuilding technique that will always let the story teller provide details to the reader in a way that feels very natural without obvious exposition.