Not sure what town you are in, but Nashua allows petitioning for age exception.

“A parent/guardian may request early admission to kindergarten or first grade. There will be no exceptions for children with birthdates beyond October 31. If the conditions are met, the child will be assessed for social, enotional and academic readiness in the summer. Parents must contact Student Services in the Masse School Administration Building to request early admission packet before June 1 of the upcoming school year.”

https://www.nashua.edu/Page/717