In june you can visit North Sikkim. The roads will be clear and at the higher region you will get snow also. But June will be little bit crowded as at that time summer vacation in west bengal will be going on. So most of the parents in west bengal visit north sikkim in june.
June is the begining of Sikkim, everything becomes green, the snow melts from high passes, so all the plaves are accessible. I have attached with this comment few packages for you, you can click on them to find more details about them. If you want to talk to someone about the packages simply fill in your details in the form and you will receive a call.