Hye Dhruv, If you have many days in leh then i would suggest go on a trek in Leh in June. You may choose between Stok Kangri trek or Spiti valley trek ( these are tough treks but if you have experience of trekking then you can try these). Hope this would help you.
You can go to tsomoriri lake . it's beautiful and to the changthang ( you will find nomads over there, the last few ), they move with their cattles and belonging. visit Kargil side . Nearby villages of Leh like Egoo, sakti , matho , stok , spituk . Go to Zanskar ( chadar trek in winter ).
Have a Happy travelling . I am from Ladakh .
Hi Dhruv, First things first, one of the sheer joys of leh is exploring Leh on foot. Try covering the monasteries two a day. If hiking is your thing. Tso Moriri to Spiti is another adventurous trek. And in my opinion Tso Moriri is actually much more beautiful than Pangong. The Tso Moriri circuit can be completed with a visit to Hanle. Now Hanle is in a league of its own. But to enjoy the real beauty of Hanle, you have to stay overnight. There are quaint homestays available. The night sky is the beauty of Hanle, the sight of the worlds highest manned observatory maintained by Indian Institute of Astrophysics. Venture a little to the wilderness and the diverse flora and fauna will amaze you. Another trek is Padum to Darcha, which happens to be the border of HP and J&K. There will be small villages en route and the best part is Ladakhi cuisine. Its amazingly delicious and very very filling. These may be some unconventional things which you can do.