Continue with the greater-than-normal drinking, but make sure what you are drinking is not alcoholic.
One tip for making this easier is to bring an empty water bottle with you and then fill it once you've passed through security. This will allow you to stay much more hydrated that relying on the slightly-larger-than-thimble-sized cups typically provided to passengers in economy. Many countries won't allow you to bring a water bottle that has liquid in it through security, but you can typically bring an empty one and fill it up prior to boarding. At least in the U.S., drinking water fountains and even dedicated fountains for filling water bottles are common in passenger terminals. In places where this is not common, you could possibly ask one of the airside restaurants to fill it with drinking water for you.
Of course, another option is to simply buy a bottle of water or other non-alcoholic beverage in the terminal after passing through security, but this option is obviously more expensive. Note also that there are some flights where neither of these options will work and any water bottles with liquid in them are confiscated or emptied immediately prior to boarding the aircraft. These are more the exception than the rule, but it happens on some routes such as China to the U.S. or Australia.
Another factor that can make a difference is what type of aircraft you're flying on. Newer aircraft like the Boeing 787 and the Airbus A350 have higher cabin air pressure and humidity than most older aircraft designs. If you're able to choose flights on those aircraft, the dehydration effect should be considerably less, though still present to some degree.