On overnight winter treks, I always bring a pillowcase that I turn into a snow-melting device later at camp. Simply pack the pillowcase with snow and hang it off a branch near (not over) the campfire. A pot below can catch the dripping water. It normally takes about 30 minutes to fill a quart of water this way so we keep the device going during our evening fire. One woodsman from Michigan I know prefers using a mosquito headnet instead of a pillowcase as his snowmelting device. Just be cautious not to get the headnet, which is made of nylon, too close to the fire or it will melt. You can also hang socks or a bandanna by the fire.
Other methods that I have used include the "snow marshmallow" where you take a large, soccer-ball sized lump of snow (like the hardpacked kind you make for a snowman), and place it on a stick anchored near the fire. Place a container under the stick, and the water should run down the stick as it melts. I have also heard of survivors using black trash bags and reflective emergency blankets with snow on top to passively melt snow in the sun.
On short dayhikes, I bring along a Nalgene water bottle that is covered from top to bottom with black duct tape to provide me with a passive solar snow-melting device. I have another (lightweight "whiskey" flask) bottle that has a 3’ loop of webbing taped on so I can wear it around my neck and inside my parka where my bodyheat converts the snow into water while I hike.
If you are going to melt snow in your cooking pot over the fire or campstove, be sure to add a little water first unless you like the taste of burnt snow.