May and June change a lot. It will finally get warmer but at the same time the rainy season comes closer every week.  Besides the Sanja matsuri, there is also the Kanda matsuri (every other year) and some smaller ones like the more local one in Iriya, the first OBon matsuri (yes in June) in Aoyama and others. Flowerwise there's mainly Hydrangea (with matsuris, though rather small ones) and Nemophila - maybe at the very beginning of May still some Wisteria.