This is how most websites work. They usually store a cookie on your browser when you log in. Then, if you open the website again either in a new tab or sometimes even on a completely different day, it can check using the cookie to see if you are already logged in.

If that's not already happening on your work website, it's likely that developers of the website haven't included the ability to save cookies to Firefox, so you will have to login manually.

To the best of my knowledge, there is no add-on that will accomplish what you are looking for.

You likely need to create a cookie allow exception with the proper protocol (https:// or http://) to make this work properly especially if you use "Delete cookies and site data when Firefox is closed".