I had initially started looking for clothes but turns out this is not what is required. By the time I had gotten this far, there was no tutorial that showed me how the cooking mechanics worked, so I didn't think to try but turns out this is how it's done. To get past the cold areas you need to cook some soup that will grant you some immunity to the cold.
All around the area before the cold section, you will see plants that are growing peppers. Pick a bunch of peppers as this is the main ingredient. To the right side of the gate into the cold area, there is a small camp with a fire. Use the torch on the ground to light up the cooking pot, having first set it alight on the fire. Take a pepper from your inventory and hold it. You will now be able to place this into the cooking pot. This will produce a meal that grants you immunity from the cold for 2:30 mins.
It's best to make a few of them in case you get lost!