Wanting to expand into another name under a parent company.
You could go about this two different ways:
1. You could register a DBA in Florida and list your LLC as the owner. This would allow your LLC to operate under multiple business names, as well as multiple websites, while using the same EIN and licenses as the parent LLC.
The main benefit of a DBA (doing business as) is that you would not need to file a separate annual report with the state.
2. You could register a new LLC in Florida and list the original LLC as the owner. This would create two separate companies, where each entity would be required to have it's own EIN and permits, such as a state sales/use tax id.