I’m currently working for a non-profit organization on a non-cap H1B visa that expired in July 2024. My employer filed for an extension before the expiration date, but the petition is still pending. I’m now working under the 240-day grace period rule. If I remain in the U.S. during this process, would another non-profit organization be able to file a new non-cap H1B petition for me within the 240-day grace period? What about after the 240-day grace period expires?

Since you have only a pending status as of now, the h-1 will have to be filed based on the receipt, this is called a bridge petition and USCIS sometimes approves these cases and sometimes sends an rfe to submit the approval, if that happens you will have to hope that you get the approval within the allotted time given for the rfe response, otherwise you can ask USCIS to approve the case for consular processing, which means they will approved without an I-94, you will have to travel outside the country and Re enter on a valid h-1visa to get a new I-94.