Behind El Mercado de la Reina, you'll find Gin Club, where you can enjoy luxuriously prepared, delicious gin and tonics (8-12€).
Next to Mercado del la Reina is the legendary Museo Chicote (Gran Vía 12), a classic bar that has always been a favourite of the famous. They are known for their cocktails, especially their mojitos (9€), Hemingway's drink of choice during the time he lived in Madrid. It's also a nice place for a coffee in the afternoon.
In summer the outdoor seating area of the Hotel de las Letras (Gran Vía 11, in front of Mercado de la Reina) is also nice. It has views of the city and a pleasant atmosphere (draft beer: 4.50€; mixed drinks: 7-9€). To get there, take the elevator to the 6th floor of the hotel and then follow the signs up to the 7th floor.
