If work music plays and clock hits the third phase of sixteen, the music will stop playing and start playing once again, just like you would start playing the same melody from the beggining.
It is the issue that appears in every DLC, not only Hamlet.
EDIT It looks like it does not have to be the beggining of the day, music can "be restarted" also later.
1. Chop the tree (or do anything else that will start the work music) before the third phase of clock will hit. 2. Keep music playing by chopping trees or doing other jobs. 3. Music restarts.