You can also find two other markets in the Florence city center - Mercato Central San Lorenzo, which is not just the leather market area outside of the church of San Lorenzo. This includes the antique building which has been restored and then divided into two sections, the original market area and a great place to go to eat with a wide selection of authentic Italian - Tuscan and even Florentine culinary treats.
Another market, which caters much more to the locals, so you will find it a bit more authentic, is the market at Sant'Ambrogio in Piazza Lorenzo Ghiberti (only open the first half of the day). You can get there on a bus, or take a nice stroll from Santa Croce. There is an indoor area with all kinds of food goodies and the outside area has a fresh fruit and veggie areas but also plants, clothing (no leather) and knick-knacks. You can also head just across the street (in Largo Pietro Annigoni) and there is also the mini antique market which opens practically every day as well. In fact, they are in the phase of remodeling it - but it is still open to the public.
The nice thing about these two mercati, both of them have an indoor area where you can escape some of the summer heat.