Fur would be a fairly traditional - and practical - liner, not least because of its warmth but also for the ease of hunting smaller game and stitching pelts together for a whole lining. Cloaks would also be hugely popular around the 1300s (and before, and also in some iteration afterward, now somewhat coming back into fashion in the modern day), because they're easy to make, easy to size (add belt!), and create a very warm bubble of air around you that coats with their sleeves are sometimes unable to mimic.

You may recall the black sheepskin cloaks worn by the Night’s Watch in Game of Thrones. (Which—Fun Fact—were made from IKEA sheepskin rugs, spurring a run on that item in that color. They’re still out.)