As we all know, that urgent feeling of thirst is your body telling you that it needs more fluids to keep working effectively. The lasting thirst or need for water which lingers after drinking what you might feel is plenty could be a result of a few different scenarios. One, and the simplest reason, could have something to do with properly hydrating well before a run.
It is often not enough for most people to simply drink a lot of water after a run or other intense exercise. Though is may depend on the particular person on how much they require to maintain adequate hydration throughout the workout, the suggested water intake for adults should be kept in mind. The fact is that other supplements are often required to maintain proper hydration, especially for those involved in endurance type activities.
The right intake of water and electrolytes, not only before your run, but throughout the day as well will make an obvious difference in reducing the constant thirst afterwards. To find the correct amount may vary some from person to person, so you'll need to experiment a little to find the right mix for you. This will be something that you can experiment with, adjusting for each run of similar distance and pace.
If you tend to be a profuse sweater, the recommended intake of electrolytes and fluids will be a bit more, as your body will need to be able to replenish what is lost. There is a lot going on in the body, even after the workout is over, so it could be something as simple as a need for an increase of continuous intake of water as well as an adequate amount of salt and electrolytes in combination.
If your activity has just recently increased from much more minimal exercise, it is even possible that your body simply requires more water to adapt to the change of added physical activity. In this case it will take your body time to adjust itself. Again, finding the right mix of water, salt and electrolyte intake is key here as well.
The weather can also be a factor. Is it warm outside? are you wearing proper apparel for the climate? Are you wearing windproofs and tights when the weather and temperature may not require it necessarily? Do you possibly overdress for cooler climate? These are things that one should ask themselves before heading out. If you retain more heat than you should, that will definitely result in increased fluid loss, which will make you more thirsty.
If you look around for answers to this question, you will find many different answers, ranging from simple fixes to somewhat extreme reasons, including the possibility of an early onset of diabetes. But, before you come to such extreme conclusions without consulting a doctor, work your way up from the simplest adjustments to your diet and fluid/electrolyte/salt intake and go from there.