On a Florida public project, a material supplier to a sub-subcontractor DOES HAVE rights against the general contractor’s bond. The same is also true on a Florida private project as to both lien and bond rights (if the private project is bonded).
Below would be the chain of command.
GC who bonded the project  ->  Sub  ->  Sub-sub  ->  Material supplier
‍
To learn more about Florida's lien laws, attend our free monthly webinars by board-certified attorneys. Click here to join