Typically, in most of Florida, once the 3 day notice time has lapsed and you have not made a payment, the next step is to submit the first part of the package to the court. The court will serve you, on behalf of the owner a 5 day summons. You can then respond to that summons if you like, but I have not seen many judges give any type of extensions on late rent.
I recommend letting the homeowner know or the management company when you will be able to pay. They can hold off on submitting that paperwork to the court and slightly delay the eviction process.
To add some info for you, if you do NOT respond to the 5 day summons, another set of paperwork is signed which includes a final judgment of default and a writ of possession. After a few days and a judges signature, the sheriffs office will be notified and paid and they can serve the writ of possession (Eviction).
Please don't use this to delay the process any longer on paying rent. I'm sure it's tough these days on BOTH parties needing the cash. Let the homeowner or prop. manager know exactly what's going on and how you plan to resolve the issue.
Best of Luck!