This is going to be an extremely unpopular answer but this is how we handle this situation now (We used to do some of the above).
Consider the scenario:
Your previous tenant is now homeless and his neighbor is providing shelter for him and his family.
Have your manager actively find some assistance for this person through the litany of organizations who provide such a thing.
The manager gets to be a hero instead of bad guy (if you value your people, this is a pretty good thing) You solve the problem. And the neighbor gets to get rid of a family who is likely causing stress on his or her family.
It's worked for us so I'll put it up here. Some answers are good, some are bad. Your choice to decide what to do in your community. I like empathy in these cases and it's not had a negative effect on our bottom line.