@menio- I would suggest looking into the way migraine type headaches start. You are explaining the exact pattern of the Trigeminal nerve.The nasal cavity is richly intervated with blood vessels/nerves. The most important/powerful nerve in the body runs right through the nose. The trigeminal nerve. “Tri” meaning 3, runs through the forhead/eye, nasal cavity, cheek and chin. Most people exhaust their efforts focusing on the nose and make visits to the ENT, to get misdiagnosed or told they don’t see anything. The trigeminal nerve is easily triggered by stress, altitude, barometric pressure/temperature changes,scents, lights, and sounds. The trigeminal nerve which is very distal from the nose, is the nerve responsible for triggering a sneeze. It is the nerve responsible for making your nose run and congest when needed. Some people with migraine headaches only feel it in their nose.
Rather than focus on the nose area, I would speak to a headache specialist. And ask them about the trigeminal nerve. If you carry stress/tension in your jaw, this nerve is already impacted. The temperature change from inside to out, is enough to trigger this nerve, if you are more sensitive.
Hope this helps.