Mistletoe??
Um... I only went duck hunting once in my life, but we didn't walk through the woods for it. Because the ducks don't hang out in the woods...they are on the water.
We got up at 3:30 am to wade through cold, knee-deep water to crouch in mud in a duck blind. At dawn, the ducks fly overhead and land in the pond or lake. At that point, you shoot at them.
According to the NY DNR webiste, the state has open season (meaning no hunting season required) only on porcupine, red squirrel, woodchuck, English sparrow, starling, rock pigeon and mock parakeet. They may be taken anytime without limit. (Maybe this is a regional/generational difference, but in the south, hunting squirrel, small birds and rabbits with a .22 was a kid's pastime. No self-respecting adult "hunter" would do it. Unless they took a kid along.)
A lot depends upon your character's personality. Is he the type of person who might be poaching, i.e. hunting game (deer, bear, pheasant, turkey) out of season? (Does he own the land? Many hunters feel that the government has no authority to tell them what they can or cannot do on their own land.)
For some outdoors types, taking a gun along on a walk in the woods is as natural as carrying a pocket knife. Maybe he carries the gun out of habit, even though he doesn't intend to kill anything? In that case, he might well carry an unloaded gun, with the shells or ammo in his pocket. Maybe he takes the gun as possible protection against encountering an angry bear?
If you really, really need a legal game season, visit or call a sporting goods store in the area and ask. But, from looking at the NY DNR website, it appears that the hunting season in New York ends by December 18, at the latest. BTW, in my experience a lot of serious hunting takes place around dawn and twilight, not so much during the middle of the day.
Here's my best shot (ha,ha): In the wild, mistletoe is a parasitic plant that grows in clumps in the tops of trees. Some people do harvest mistletoe by shooting at it with a .22, to sever the clumps from the tree. (Please don't try this at home...it's extremely unsafe. You don't know where that bullet is going to end up.)
For many woodsmen, fresh mistletoe is a Christmas traditon. Plus, it's a really good excuse to be in the woods, which is where they want to be anyway.