Step 1:  
>! Let's get the wood skewer. <s>Since the screws are flat heads, I'll assume that the rim of the pot just fits into the head of the screws, and thus you can unscrew them using the pot as a makeshift driver.</s>  
>! Okay, well I guess we still have a few options:  
>! Keep in mind, we technically walked through the door with the makeshift saw from the previous puzzle. (Nothing noted that we dropped it, and I don't see why we, as a puzzle solving survivor, would).  
>! So let's try a different couple of things and see what works. If the first attempt doesn't work/isn't allowed, move on to the next.  
>! Attempt 1: Use one of the edges of the glass blade to unscrew the screws.  
>! Attempt 2: The now cut wood that was previously locking the door we just came through could be sharpened or shaved using the glass saw we have. Refine the wood down to a point that will fit the screws, and then unscrew them.  
>! Attempt 3: I guess if all else fails, @Yessoan came up with a pretty interesting idea we could try. I'm willing to use this mainly because I have other ideas and because their water idea and mine differ.  

Step 2:
>! We need water. For drinking and other reasons... So, over to the tap we go. Put the pot under the tap, along with you coat ontop of the pot (wool side facing up towards the faucet.) Turn the tap on low and let water go through the coat into the pot. (Or if it can't, wring the coat out into the pot every now and then.) This will reduce the salt content to help give you drinkable water. Also, we'll need some for the next step.

Step 3:  
>! Boil the bent wood skewer in the pot of water using the fire. Eventually you can remove the pot and straighten the wood back out. (Use your wool clothes as oven mits if needed). 

Step 4:
>! Pick the lock with the now straight wood skewer. (Or, if needed, bent into whatever shape helps)