Does MySQL get the old write time?

I have a table in my database that contains all the users for my application. Unfortunately, when I launched the application, I didn’t think of including a column that tracked the time at which a particular user was logged in, and now I'm sorry for what I have (a bad idea, yes indeed).

Is there any good luck case, the way MySQL keeps track of when a particular record is inserted (for example, in record metadata?) And allows me to grab it and insert it into a new highlighted column for that purpose?

I am running the cPanel shared host, so I doubt that I have access to MySQL logs.

Thank you for your time.