I previously mentioned my connection to Jews whose roots are in the community of Djerba and tonight I learned about a custom of theirs of which I had previously been unaware. Apparently, Jews from Djerba, along with some from Yemen, burn their hametz on the evening of the 14th of Nisan after they do bedikat hametz. Some of them burn hametz again on the day of the 14th, while others only burn it on the night of the 14th. On the website of Yeshivat Kissei Rachamim there is a responsum with some sources that describe this custom. The following book gives a good overview of the practice.
חג כשר ושמח.