Even on a tight schedule, Doge's Palace 'Secret Itineraries' Tour earns the hours.
Allow ~75 min tour + palace.
This small-group guided tour takes you behind the public route into the cramped administrative offices, the interrogation/torture chamber, the leaded 'Piombi' prison cells and Casanova's escape route — the dark machinery of the Republic the standard ticket never shows. The catch: it must be pre-booked, it's a specific timed-language slot, and you still want time for the main palace afterward (which it includes).