If she was familiar with the information that was on USS Defiant - and there's every reason to think she would be - then she may have noticed that things aren't tracking quite right with what was said there for the last few years, and tracked the apparent source of the differences back to CONTROL, which she probably would have been familiar with a version of in her home universe. Section 31 only thinks *they* recruited *her*, when really, she got herself associated with them in trying to get to the bottom of a mystery.
Good call. I think something to do with the Defiant database is even more likely given this cryptic exchange:
Burnham: So you're telling me this out of the goodness of your heart?
Georgiou: No, the goodness of yours. I know so much more about you than you can imagine. But that's for another time.
There could be other reasons, but it makes sense that part of why Georgiou is so good at her job is because she has foreknowledge. I also find it unlikely that they're going to do something to make her less sympathetic given that Michelle Yeoh is getting her own show, so I'm on board with the idea that she's out there investigating things based on the database and staying a step ahead of everyone else. (Rather than masterminding Control's descent into murderous tyranny, for instance.)