I'm working for company A. Company A is outsourcing me to company B, and company B is outsourcing me to the company C, where I work on a project for a client.
I'm leaving the company A soon, and my boss does not want me to talk about this in company C. Instead, he wants me to continue working as usual without mentioning anything, and then he will somehow announce (or company B will) to company C that I'm on a sudden and long sick leave and someone else will continue my work.
However, everyone in C will know that I've just changed jobs, because I want to update my public LinkedIn profile when I leave A. This is a very niche field I'm working in, so I will certainly work with some of the people in company C again in the future. It's not a big team, but I'd like to say a proper goodbye to the people I've been working with, which is a professional behavior for me.
If I leave without a word, and they will see that I've changed jobs, and they are told that I'm sick, this could be suspicious for some people. On the other hand, I'm still employed at company A and I don't want to part on bad terms with them by doing exactly what my boss tells me not to do.
What should I do to be professional and not risk my reputation? Maybe I should talk to my boss and try to convince him that people at C will learn that I left anyway?