When you say "just after" you mean the Driver's side, and if so, you are saying that the needle stays between the driver's side door and the "L" on your gauge. That is not normal, just to the passenger side would be normal, but what is important is where does the needle go once the car is warmed up and running. If the needle still stays down there, try to change the oil pressure switch, it is located at the back of the Block. If you are still getting a low reading ,do an oil pressure check. How is the car driving? Do you have a 429,428, or a 390?