93 Chrysler 5th avenue/Newyorker dies?

Ok so I recently went car shopping and bought this car from a family memeber. Its in good condition on the inside and out its been a garage kept car sense birth. New rebuilt transmission has been put in along with many other things you would do to keep a car going oil changes every 3000m. Here's the problem for some reason every 100 miles you have to re fuel the car back to the very top or it dies. We have had the gas tank taken off several times the gas cap has been changed computer checked out. But for some reason if it is not always full of gas it will die later on and the only way to get it to come back to life is to put more gas in or stick a pen or pencil down in the gas tank opening and push the flap down to let air in. Then get gas to be safe. Its really annoying considering how well kept this car has been. I feel its something that can be fixed easy if I only new what it was so any help would be great thanks.