Like May, June is worth considering as you can still explore and enjoy the bulk of the country with temperatures in the high 20s°C and expect only intermittent afternoon showers. The main incentive is the ability to advantage of lower airfares and hotel prices whilst avoiding the crowds.
You do have to be careful when choosing your beach destination. Koh Chang (close to Cambodia) is wet and the west coast will be experiencing increased rainfall. For the best conditions head east to Hua Hin, Cha Am or Khanom on the mainland and off shore on the islands of Koh Samui and Koh Phangan. Perfect dive conditions can be enjoyed around these islands.