2004 Lincoln Aviator Won't Start intermittently

I have a 2004 Lincoln Aviator that will start fine, but if I turn it off and try to immediately restart it (like after pumping gas) it will not crank. The dash lights come on, but it will not attempt to turn over. I just replaced the battery, but that did not fix the problem. If I let it sit for a few hours or wait until the next day, it cranks right up. Any ideas?