This is a very interesting prompt. In order to do justice to it, you need to be able to put yourself in Blanche's shoes. She's just come out of the mental asylum, so naturally, she's had a pretty rough time of it. Being sent to a psychiatric facility would be a stressful ordeal for most people, especially in Blanche's day, when the treatment of mental illness was nowhere near as advanced as it is today. That said, if the authorities have deemed her fit for release, chances are that her mental health has improved.
In writing an extension to A Streetcar Named Desire, you could perhaps develop the theme of Blanche's loneliness. She was lonely before she was sent to the asylum, but now, upon release, she doesn't have anyone to turn to. She can't very well go back to her sister; she tried that once, and it turned out to be a complete disaster. With no family, no friends, and her reputation in ruins, Blanche is about as lonely as it's possible for someone to get.
However, it's possible that Blanche made a friend while she was inside the asylum—either another patient or a member of staff. Or perhaps someone wrote to her while she was institutionalized, offering her a chance to start over once she was released. Drawing on this possibility allows you to use your imagination and construct an interesting story.