I have recently got a new job in September. How soon is to soon to add it to my resume?
If by resume you also mean your LinkedIn and other online accounts, then this has been debated for a while.
Decades ago nobody saw your resume unless you were looking for a job, or if the current company wanted to use your resume to win new work. Now with the online resumes everybody can see them as soon as you update it. Even you current manager.
Do you enter the new job over the weekend between the jobs? Right after you start? After a few months. But if you wait too long to put something, the more worried your new manager gets. I have even know people who waited a year, to make sure the old company wouldn't know where they now work. The figured people would stop looking after a couple of months.
During those first few weeks/months your paragraph about the position will only be based on the job description, it will be short of details and accomplishments.
You have to balance what you need to advertise your skills and accomplishments, and making it easier for people to find you.