Non-Profit institutions are EXEMPT from the H-1B Cap, which means you will not have to worry about rushing to file. Your petitioner-employer can file right now. They may even can opt for expedited processing - your petition could be approved within ten (10) business days. Please call for a consultation.
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Aliens are exempt from the cap on an individual basis among others if: are resuming cap subject employment for which the maximum 6 years has not been used. see INA Section 214.(g)(7); 63 Fed. Reg. 25870-71 )May 11, 1998); USCIS AFM Section 31.3(g)(14) (remainder time). It is unclear whether the portability rule applies to a change from a cap-exempt employment to a non-exempt employment. But if you are seeking cap-subject employment without having ceased to be employed in cap-exempt employment, see INA Section 214.(g)(6); USCIS AFM Section 31.1(g)(13)(D) it "may" work but would require additional research. Perhaps other colleagues can shed more light onto your question. Good luck.
My colleague is correct in that nonprofits’ are cap exempt. As such you would not have to wait until October. I would recommend hiring an attorney to make sure filing is done properly and expeditiously, in your interests. You can begin working at new job when the paperwork is filed with USCIS.
This is a very cursory review of your situation based on the limited facts presented and should not be construed as legal advise. (954) 889-2244