I have constant head pressure as well. I'm still not sure what the cause is, however I have found apple cider vinegar to be an absolute miracle and relieves so much of the pressure that I do not need aspirin every day.
I have also found oil of oregano to be greatly helpful as well. Just want to share with everyone I possibly can who also has head pressure and headaches. i take 2 tablespoons in juice and 2-3 drops of oil of oregano also in juice every morning and evening. I find the liquids to be more effective than the pill forms.
They are both very strong, so you will likely need a "chaser". You can also warm them both up in a cup of water then mix with tea, cinnamon and honey. If you don't have enough time for the warm version, go the simple juice route.
I hope this provides someone with as much relief as it has for me.