If he won't vacate willingly you will need to go through the legal process. If there is a written lease, you will have to let him live out the terms of the lease and then not renew it. If there is not a written lease, then legally he has a month to month tenancy and you can evict him with a 30-day notice.
If you are able to give him a 30-day notice I would do so in writing and serve him immediately. This will speed things up if you have to take him to court. You may sue him for eviction if the 30 day notice expires and he has not vacated the property.
This answer is intended as informational only, and does not constitute legal advice or form an attorney-client relationship between us.
If you have a divorce judgment/MSA, look to that first - if the issue is addressed, follow the terms. If it's the ex that's failing to follow the terms, file a petition for rule. But before doing anything, see a family law attorney in your area for specific guidance.
If I understand correctly, you were both awarded the marital former property in your divorce but your ex has since executed a quit claim deed. I have never heard of an exit contract but your suspicions about his intending to stay for free seem plausible. Therefore, even though you wish to avoid this, you will probably have to go through the eviction process. You and not your ex now own the property along with the bank. The most efficient way to handle your hanger-on is to consult with an experienced eviction attorney. You should discuss whether you are able to begin the process now or you must wait until you close on the refinancing. In any event, based upon your description of your ex, you will need legal assistance to help you get rid of him. Therefore, the sooner you start the sooner will be able to get him out of there.
In advance? Without reviewing the decree it's really impossible to say for sure. The decree should have defined what his continuing ownership AND occupancy rights and obligations are, which might create a tenancy you can deal with in eviction court, but if there is any ambiguity in it then your only recourse might be back to divorce court. For example if the decree obligated him to sign some kind of additional contract (I've never heard of an "exit contract") in the nature of a rental agreement and he doesn't your recourse could be to take him back to divorce court to force him to sign it. Otherwise seems like you're in a catch-22 and if you insist on some written arrangement for his continued occupancy after he gives you the deed (which he apparently is NOT obligated to do), he could simply withdraw the deed.