For this kind of region there are two types of travel time . One is ICE SEASON & another one is FLOWERING SEASON . So it's totally depends on you . For ice the best time is JAN to FEB & for flowering season APR to MAY . And for your concern I think June is not kinda good season for traveling around Hilly areas . Cause as far as I know , it's basically a rainy season and flood can happen in any time , plus the view is generally busted during rainy season. I hope this will solve your question 🙂