The new version of Facebook Messenger show a chat head when new messages arrived.
When closed, I don't know how to show the chat head unless waiting for the next message to come
A free app called Notification Shortcuts gives you the ability to set up a permanent notification that will open an app you specify when you select the notification.
You can set up multiple shortcuts and can even set up toggles for wifi, bluetooth, etc, if you want.
The downside to this is that when you receive a message, you'll have a second notification from the chat app itself. But since the facebook chat app lacks an option to set up a permanent notification, this 3rd-party app is the next best solution.