I am zoned in a school that is not a school I would be interested in putting my children in. Are there schools that are open and accept kids from other districts in Manhattan?
Probably not. District 2 and 3 have some "unzoned" schools and magnet programs that are filled by lottery, but space is at such a premium that schools are not likely to accept children from outside of their zone.
You can also have your child tested for gifted and talented. Any incoming kindergartener scoring at least a 90 is guaranteed a seat in a gifted and talented program.