After Orsanmichele, I suggest visiting another loggia nearby that's called the Loggia del Mercato Nuovo, or New Market Loggia, or more popularly, Loggia del Porcellino, or Piglet Loggia.
But first take a look at the nearby Arte della Lana, or Wool Trade palace, which is quite ancient but has been significantly restored and is connected to the upper floor of Orsanmichele by a bridge. The palace was the headquarters of the mighty Wool Trade, which was the trade association for wool mill workers. Note also the 14th-century tabernacle on the corner of the building.
From the names not only of the palaces but also the streets, you've probably already realized that you're standing in a part of the city that's closely linked to the history of medieval corporations which gathered here to defend merchants and artisans, but also doctors and notaries, and even barbers.
Via Calimala, for example, takes its name from one of the oldest and most powerful corporations of luxury fabric manufacturers and merchants.
But now let's talk about the New Market Loggia. It was built in the middle of the 16th century by Cosimo I de' Medici for the sale of silk and precious objects, while in the 1800s the famous Florentine straw hats boomed here. It was part of the series of specialized covered markets that had been designed by Vasari to favor the rational organization of the city's trade system. Unfortunately it will be hard to find something original among the stalls that are always popular with tourists today: it will all likely be a more or less kitsch souvenir, or a more or less genuinely Florentine leather object.
Don't miss the bronze Boar affectionately called the "porcellino", or piglet, which decorates the fountain on the corner of via Calimaruzza near Ponte Vecchio. Also in this case the sculpture is a bronze copy of the seventeenth century original by Pietro Tacca, which you can admire sheltered from the weather in the Stefano Bardini Museum.
FUN FACT: you probably noticed that the boar's nose is much shinier than the rest of him. According to tradition, rubbing it brings luck, but that's not enough: you also have to put a coin in its mouth in such a way that it passes the grate where the water falls. Otherwise you won't get the good luck!