I am in L-Permit which will expire in Nov, 21. My contract (project) is going to get over in Mid june 21.
Can I stay in Switzerland after mid June 21, even if my project get over.

key question: is your permit linked to the employer? And by project/contract being over, you mean you'll no longer be engaged by them?

what passport do you hold?

Yes, Permit is linked to the employer. Once project will get over, logically Swiss employment will employer will get over, however I will be employed with employer from India.
Non-EU passport (India).

Then your permit ends with the end of the contract.

what are your guesses as to whether you can?

You mean to say that there is no grace period. Last working day with employer is the end of permit irrespective of permit end day.

It works like this. Once your Swiss contract is over, the employer must report this to the immigration office. If you're still registered in Switzerland they'll inform you that your permit is now void and set you a deadline by which you need to leave Switzerland. It's usually about a month. They're usually reasonable and can extend this if you have real extenuating circumstances but don't expect to be able to stay until November if your contract ends in June.

My Spouse is getting permission to work on Dependent L-permit. In this case, can I stay once my swiss contract get over and my spouse can continue working on L-dependent till Nov end?
Migration office informed spouse employer that they can initiate new application for employee in Sep'21 considering L-permit expire in Nov'21 and proof that current primary L- permit applicant is not currently in work.
I am not sure how this will work as grace period is normally for 1 month.

so your spouse is getting some work depending on your permit which will cease soon?

I suppose that does not matter because your spouse will get a permit linked to a contract.

Can you drip-feed the information a little quicker so as to make this more exciting for us not in the know?

No. Their permit and right to work is based on yours. You leave job and loose permit, so do they.

I'm not sure why the migration office even said they can initiate a new application because the law is quite clear.

"Family members of a holder of a short-term residence permit (Article 26 OASA) do not enjoy the right to exercise a lucrative activity. The exercise of a lucrative activity is subject to authorization. Upon admission, a request from the employer respecting the remuneration and working conditions in use in the locality and branch must have been submitted. In addition, members of the family of a holder of a short-stay permit (Article 26 OASA) must have professional qualifications (personal qualifications, Article 23 LEtr).

The possibility of engaging in a gainful activity for family members is related, according to art. 26 and 27 OASA, for the duration of the authorization of the person who has benefited from family reunification. If the spouse's residence permit is not renewed, his family members can not claim a right to continue their gainful activity (Article 6 (2) OASA)."

Your permit expires in November and so does hers.

Only thing I can think of is if she has exceptional/niche skills they may allow the potential new employer to apply for an independent one effectively, which would make you her dependent. Sounds strange though and would be very unusual if that's the case.