Saw this on one website.
Small game season starts in early October; large game/deer season begins around Thanksgiving, when hunters have exclusive use of the Forest into December. Turkey season begins May 1.
Hunting for all species is allowed in season; however, vehicle access to the area is prohibited. Hunters must park in one of the peripheral parking areas and access the area on foot or bicycle.
I believe bow hunting has begun as of the 1st, but the park is still open. Stewart is only closed to non-hunters during the 'regular' hunting season, which I have found to be 11/21 - 12/13. I'm assuming that's the time frame to stay out.