Provide more info in chat about device type, location, browser type, source, etc.
Having the ability to see
breadcrumb trail of viewed pages in this and previous sessions
IP Geo location
device type - mobile, tablet, desktop
referral source / organic / ppc, social etc
Are a few great ways to help the chat rep have a much better understanding of the current visitor so they can better provide quicker help. For example, if you know certain parts of your site may not be ideal on mobile and the person is having an issue... or if they are asking for a price... by seeing this other info you can have a pretty good idea of what product they were looking at and hoping to find pricing on.
Also as the user continues to click around the site, insert that URL in the chat timeline so you can know where they are continuing to click around to as you talk with them.