My problem was described here. We need to save inforamtion about "Viewed Pages" and "Current Page" in the LiveChatTranscript record. Maybe somebody know any solution how to save this data?
During my investigation Salesforce Console Integration Toolkit Developer Guide I discover that it possible to get list of all Viewd Pages by user through the method getDetailsByChatKey(). Viewd Pages that we can see on UI in the LiveChatTranscipt record when status is "In progress" are stored in breadcrumbs array of details Object that return after we invoke this method.