I'm not really on board with this approach. For one, you don't really have a full understanding to whom the candidate wants to disclose that they are on the market. Their current employer is a no-no, but other projects might be ongoing part-time roles, etc. It might not be as simple as just skipping the most recent project.
Second, you don't know the circumstances and the person might just be a jerk or be bitter solely for the reason that they got left in the dust.
Third, IMO it reflects poorly on the candidate that they did not notify the reference before he/she is contacted. Not many people like to get caught off guard like that.
Yes, references are hand picked to be good. If the candidate has 3+ references that all give glowing reviews, does it really matter that you don't get the story from absolutely everyone? If your answer is yes, then at least do the candidate the courtesy of asking permission so they can give you any necessary background info.