Visiting spiti is really a nice option in June. you can go for a short trek to triund in Himachal. in Uttarakhand you can visit nainital , bhimtal etc.
If you're based out of South India I would suggest you to go to Chikmangaluru as many trekking spots. Also Mandalpati in Coorg Ooty also scenic trekking spots.
Mecloadganj!! you will love the nature and weather. 5 to 6 hours trekking with some amazing cool cafes, and try to stay there for a night seriously you will love it.
Hi there are lots of places to visit in Himachal, Kinnaur, kasol and Jibhi are few to name. I have attached with this comment few options 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.
Most places in Himachal will offer you brilliant view . You can stay at Mcleodganj , Dalhousie , Khajjiar etc
Even Darjeeling will offer a stunning view of the mountains But be mindful of the rains . check weather before you plan