Thanks for bringing this up! I have the same problem now and then and I am SURE it's not just you and me. The last month is a very vulnerable period for a landlord. Because lets say the damage deposit is one months rent, which is typical. The tenant tells you they will move out on the fist of next month. And they just don't pay the last month. By the time you are pretty sure they are not going to pay you are two weeks in (because just like you said, they were by then already consistently paying late). Now what are you going to do; get an expensive eviction process started while they are going to leave in two weeks anyway? On top of that you risk pissing them off to the point of damaging your property. So yo are just left to hope they will not leave damage and cleanup that is disproportionate to the time they had been renting. Lets say there is a couple of hundred worth of damage or cleanup. Good luck finding someone what wants to go and try to collect a "small" amount like that. And how much time do you really want to spend chasing something you will never see again. And they people don't care because they were system gamers with nothing to loose to begin with.
I have not found a good solution for this myself. In theory the best solution would be to never get someone like that in your property to begin with.:)) So easy to say. Care full screening, helps but I found it to be not fool proof particularly because a certain type of asset class attracts a certain type of tenant. So to some extend you have to pick between bad, not so good and worse:).
I always try to get a double or even triple deposit if I can before they move in if there is any reason there might be an issue. (check your local laws). If they turn out to be great after a year or so I don't mind refunding some of that either.
Depending on easy it is to find a new tenant quickly (and if they are on a month to month) you might consider asking them to leave right away if they start paying late. I have tried to work with people* who were "starting to get late" and unfortunately I have so far never seen that end with the tenant being back on track.
*The reasons I worked with them were rather selfish I must admit...:) For example if they live in a unit that I can NOT market without remodeling and I do not have the money or time to remodel, I find it is in my interest to try to get them to pay for another while, even if in the end I only see half of the rent. As long as they are not creating issues for other tenants this is still better than no rent for an empty unit me paying the utility bills on top of that. Once I am flush again and am ready to tackle the rehab I can always turn up the heat.
I am really hoping that someone else here's going to come up with a great way to navigate this!:)