When Tony Franciosa visits Jean Simmons at her school, while he is outside the classroom, you can see a drawing on the chalkboard of a movie camera. (Approximately 25 minute mark).
Once inside the classroom, the chalkboard the usual writings you'd see in a teaching setting.
No reason is given for Jean Simmons taking a second job. She is a teacher, so is earning a decent salary, and she lives very modestly, so she does not need money, and there is no mention of her needing to support anyone or to save up for anything. It is puzzling why she would take a night job that, one would think, would make her too tired to do her teaching job.