The Palazzo Vecchio is now open until midnight almost every day of the week (which is still not noted on their official website), and, all the doors (E,N,S, & W) are also open during the day. Also gone are the metal detectors. What it makes for is a much nicer experience - you can walk into and through the building and see a lot more of the ground floor without paying anything. Or you can buy a ticket from the ticket office (full price is €6) and mount the stairs to see the whole museum. I really want to go in at night but I am trying to wait until the weather cools a bit!
Verrocchio's putto fountain sculpture in the first courtyard: