The phrase "already getting to know" is correct and usable in written English.
It can be used when discussing the process of becoming familiar with someone or something over time.
Example: "We have been spending time together, and I feel like we are already getting to know each other better."
Alternatives: "starting to understand" or "beginning to familiarize".
"You see," she says to Ashima, "he's already getting to know you".
"Geoff's been over here already getting to know everyone and seems a great lad," said assistant boss David Kemp.
Giving credit where credit is due, I overheard one teen viewer mention that she was already getting to know the characters - to "forget that they were fat".
"Except for the one neighbor that we had already gotten to know, nobody came," Ms. Barlow said.
Alone among them, Mr. Dobson had already gotten to know Mr. Graf during the organizer's brief months in the city.
When Reeves met Carson in Columbus, in the summer of 1935, he had been in the infantry at Fort Benning for four years and had already got to know Carson's mother.
I had already gotten to know Ms. Siegel well enough to understand that her choice of woodworking as a career was, as she freely admits, a rebellion against her upbringing on upper Fifth Avenue.
After that hour, the rats were exposed to a new object and the one they'd already gotten to know; if the rats remembered, they spent less time checking out the familiar object.
Companies that are CRM-centric (e.g. hotels, airlines, retailers) that already get to know their customers as people are pretty comfortable with the 1 1 interaction on social.
Keiko and I are both very proud of it, we've been nurturing that material for ten years, which is very unusual for a recording project anyway, to have the recording take place so far down the line, after you've already gotten to know the material.
At this point, you've probably already gotten to know her pretty well.
