Problems starting my 90 Plymouth Grand Voyager LE First Time in the Morning

I have a 90 Plymouth Grand Voyager LE van that is giving me difficulty starting 1st thing in the morning. At first it would make no sound at all when I turned the key on. After 2 or 3 tries it usually would start and would always start when I went for lunch and at the end of the day. I had the battery checked and they said it was bad so I purchased a new battery. For 2 straight days it started first time in the morning but the 3rd day it went back to not starting on the first crank. I went to Auto Zone and had the alternator checked and they said it was fine. Now in the morning it will make a click the first time and then the 2nd time act like it is going to start and usually the 3rd time it will start. I don't know what to do next. I know that possibly the starter may be the problem or a bad solenoid but it just puzzels me that the only time it gives me these problems are after it has set all night. Any suggestions?