To piggyback on what Ms. Vivian is saying, a good way to approach is to simply continue the conversation you had in public, in the message. If it’s interesting and not blatantly transparent, then most girls will respond even if they sense you have an agenda.
I would not try to start a new conversation about a new topic as a first private message. I’ve tried that before and while she responded, it wasn’t a pleasant conversation. Even though she didn’t think I was being creepy or weird, there was a lot of misunderstandings and I think the jarring aspect of new conversation + new venue was to blame.