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.
The narrator of Treasure Island (1883) similarly says "I take up my pen in the year of grace 17--", and the letters framing Frankenstein (1818) are also dated with the year "17—". Take it up with Trollope, Stevenson, and Shelley.
– Zan700 Jan 15 '21 at 23:24There was a style - somewhen mid-Victorian? - using dashes to obscure not only dates but names of places and even characters; never needed but not uncommon.
If you're really setting a story in a year you yourself don't know, good luck with it… the more so since "the 2030s" is almost upon us. If you think 2030 and 2039 won't be as different as chalk and cheese, doubly good luck!
Why not write it, then think about date styles?
– Robbie Goodwin Jan 21 '21 at 21:33Try reading The Quincunx, seen by some as “an absolutely stunning literary achievement. Much more than that, it's a great read.”
As that link suggests, it is rather Dickensian yet there is no clear reference anywhere its huge length that would date it within 100 years either way.
– Robbie Goodwin Mar 07 '21 at 20:41It's not about greater achievement; simply that Q having got there with neither any reference to dates, frames or periods, nor any obvious textual clues, so can anyone else, including you.
If you're happy with "Los Angeles in the fourth decade of the twenty-first century", what's any of this been about?
– Robbie Goodwin Mar 07 '21 at 22:56Is it not obvious, it would never be reasonable for anyone to explain how each and every sentence contained no clues?
Read the book, then discuss it in general and most particularly, how the author managed to write so many hundreds of pages without ever once mentioning anything to do with a date.
– Robbie Goodwin Mar 07 '21 at 23:17"No one would have believed in the last years of the nineteenth century that this world was being watched keenly and closely by intelligences greater than man's and yet as mortal as his own…"
The only point of that apparently crucial sentence was to establish a time frame… yet establishing that time-frame in no way helped either the plot or the telling of the story.
– Robbie Goodwin Mar 23 '21 at 20:24