I overheard a phone call by an employee while passing by, who was on call with a contractor, saying negative things about the company and her job.
How should I deal with something like this? I would like to address the issue without her presuming that I was somehow snooping on the conversation on purpose (which wasn't the case). It also concerns me that her dissatisfaction will create problems in the team overall.
Talk to her.
If she's dissatisfied, it should be showing up in other areas; timekeeping, motivation; quality of work, interaction with others, which you could use to start the conversation - although the bad mouthing alone is a good enough start point. Starting with any of these is going to make her defensive, and feel like the start of disciplinary processes. Is that what you want?
The best alternative is just to ask her "how's it going?" and take the conversation from there.
Side note to help this repeating: Consider why you didn't spot this already, and why she hasn't come to you voluntarily.