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The phrase "had already reached" is correct and commonly used in written English.
It is typically used in past tense to show that something had happened before another event in the past. Example: By the time we arrived at the beach, the sun had already reached its peak in the sky.
Exact(60)
The news had already reached his family.
By June it had already reached 1.2%.
The Taylors had already reached Kansas City.
They had already reached a verdict however: guilty.
After one lap, his lead had already reached a second.
In doubles, they had already reached the No. 1 ranking.
By early summer 2010, the temperature had already reached 100 degrees in Cincinnati.
I was also dimly aware of someone else who had already reached him.
But unlike the other commuters on board, the three had already reached their job.
Henry was late; the discussion had already reached its terminal stage.
Van Artevelde had already reached middle age when he began to take part in public affairs.
More suggestions(18)
had already attached
had initially reached
had indeed reached
had now reached
had yet reached
had really reached
had once reached
had ever reached
had previously reached
had again reached
had just reached
had also reached
had even reached
had recently reached
had already reach
had already been attained
had already been received
had already been learned
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com