We're a small, special library with several information specialists. One of the ways we've improved use of our chat is to include the widget in our No Results database and 404 pages. We saw an increase of about 15% when we did that.
The canned responses in the web chat helped us greatly in being able to respond very quickly to incoming chats. All I have to type is He and it pulls up our standard "hello!..." intro. Hit enter and you've started your session.
You could edit the widget with some text at the top along the lines of "An information specialist is on the way! Please give us up to 1 minute to respond. We appreciate your patience!" along the lines of the hack to add a link to a feedback form (am I remembering that right?).
Something for the feature queue, perhaps, is to have the option for an intro script that pops in when the user starts a chat/opens a chat window. "Thank you for contacting us! Please feel free to type your question or additional information while the next available librarian takes your call." You could add your "We may take up to 1 minute to respond, your patience is appreciated" there.
---Jess