The blanket you had transfigured did go through the cage. While magic couldn't get past it, the physical could. However, wrapping the blanket around your friend was a different matter entirely. The bars were sure to serve as an obstacle.

Using one hand, you were able to drape the blanket over one of Sam's shoulders, providing a slight source of warmth for her. It wasn't enough to protect her from the cold arctic temperature, however. The sooner you get her out of the cage, the better.

The small Ravenclaw's hand definitely is freezing to the touch. Though you have warming charms casted on you, a slight chill can be felt and you noticed that her hand is slowly starting to turn a blue hue. When you start gently prying at the object in her hand, your assumption of it being a piece of parchment is confirmed. What could it possibly say?

After you remove your arm from within the cage and unrolling the scroll, you will see that numbers needing decoding have been written on it:

285 6995 19751