This is strictly the fact that I have proof she is working two jobs that are in direct competition with her job.
So AFAIU, she is working two jobs additional jobs when, by contract and/or expectation of HR she should be only doing none without registering with HR. Potentially she did this without anybody knowing, so judging may is none of your business. But lets assume she did not, then the following applies:
I do not understand how this "does not hurt the company". It is central to the expectations of an employer where you have an employment that you are supposed not to work side jobs in secret. Even if it does not hurt the company currently every day in a measurable way, it still has the potential to do so, and, depending on her behavior in the other jobs actually already is (without you knowing). This person would be self-centered and maybe lying to cover up the fact that she works multiple jobs. Such behavior undermines everything which you need in a team. Besides she have proven one time (together with you old boss) that she does not hesitate to act against the company interest.
As i wrote before, you don't know what HR knows or not. I don't know the country you are in, so i don't know their viewpoint and the laws on it. You could do one of the following:
- The most straightforward way would be when you find a real problem. E.g. orders being processed late or constant difficulty in arranging a meeting since she has strong time restrictions from her other jobs. If that is the case, go to you boss (assuming she is not sleeping with him again) and state this specific thing. If he says something like "I dont understand why colleague doesn't have time", consider it an invitation to speak out.
- In Germany: go to the working council and ask for advice (without telling the colleagues name)
- Go to HR or you boss, talk to them, first asking without telling the colleagues name if they would like to be informed in such a situation. "It came to my attention that a colleague is potentially working additional jobs and i wonder if I can or should tell you more specific about this suspicion confidentially" - never state it as a fact, it is not your task to establish if it is a fact.
I personally see this less as an ethical problem (she exhibited behaviors which hurt companies in the long run), but more one where you could get burned if it turns out that she registered the jobs.