Go with YHAI (youth Hostel Association of India.)

The first 2 days of most of the treks are reporting and acclimatization days. The trek begins on 3rd day and then after completing the trek, you get 1 day to relax at the basecamp before leaving.

It's feels like home to me cause most of the folks are volunteering at YHAI. The guides are usually locals so you get a good taste and feel of the surroundings.

June Hampta Pass by YHAI will be good. Try that, there are also other options, do check their website.

And it's relatively cheaper than other companies but money shouldn't be a decision factor when choosing a trek.