Bear canister rules are often relaxed in the winter. However, this will depend on where you are camping. On the east coast, in the Adirondacks, the rule is:
NYSDEC Regulation Requires The Use of Bear Resistant Canisters by
Overnight Users in The Eastern High Peaks Wilderness Between April 1
And November 30. NYSDEC encourages campers to use bear resistant
canisters throughout the Adirondack and Catskill backcountry.
On the other hand, out west bear cans are often required year round.
Obviously, you'll want to check the rules for the specific park you are going to be in, but as a more general guideline, if the temperature is generally staying below freezing there shouldn't be much bear activity. Combined with the general bear population this will determine if bears are likely to be a problem.
Another thing to consider though, is that hanging your food protects it from more than just bears (eg mice). While the bears might not be a problem, mice don't hibernate.