There might be something where you sleep that triggers your runny nose, and then it takes awhile to recover a few hours after you wake up. I might even try a different pillow and see what happens. Otherwise, elements of your diet could come into play such as garlic, turmeric, etc.. Eat more salads/leafy vegetables.
It’s probably due to the fact that in Winter I sleep with a humidifier going and when I get up and start moving around the drier downstairs, it’s off to the races.