My flight from MIA-HAV was a charter through SkyKing. They fly daily service to HAV, but it is coordinated through a few different travel agencies. Marazul Charters comes to mind as one of them.
So, it would not really work to fly through MIA to go to HAV. Other places you could look would be via Cancun or one of the direct flights to Europe (Condor from FRA, KLM from AMS, AF from CDG, Cubana from ORY).
Here is the link to the US charters to Cuba
http://cubatrips.org/charter-flights-to-cuba.html