First thing I would check at this point is the fuel pressure after you turn off the vehicles engine. The fuel pressure should hold for a little while, if the fuel pressure is dropping fast after the fuel pump shuts down, then the check valve in the fuel pump could be leaking and after the vehicle sits the fuel line could be empty and it would take a few seconds to fill the fuel line back up and have fuel at the fuel rail. Earl is correct in stating that the fuel relay and PCM signal to energize the relay are good, since you do not have the oil switch backup circuit anymore.
when the fuel pump check valve leaks, it is a leak back into the fuel tank, not an external leak that you would smell or see on the floor. Just wanted to make sure there is no confusion.