I believe the fashionable Flaneur in top hat at the left edge of Manet’s painting is the poet Charles Baudelaire. According to Alan Bowness (Poetry and Painting: Baudelaire, Mallarmé, Appolinaire and their Painter Friends, Clarendon Press, Oxford, 1994/ quoted at a&a):
It was Baudelaire’s friendship that gave Manet the encouragement to plunge into the unknown to find the new, and in doing so to become the true painter of modern life. The Music in the Tuileries is a new kind of painting… Baudelaire appears at the extreme left, fashionably dressed, talking to Gautier. In the final paragraph of his 1845 ‘Salon’ Baudelaire had stated that the true painter for whom we are waiting would be the one who could find an epic quality in contemporary life and make us understand ‘combien nous sommes grands et poetique dans nos cravates et nos bottes verries’. This is exactly what Manet has achieved.