Neal, This a common problem with older models. What you got is resistance from the battery to the S terminal. And the problem will get worst. By the time the voltage reach to the S terminal it has dropped at a point where the solenoid will not engage. Adding the relay as I suggest.
It will prevent any voltage drops because the contacts is connected between the battery post and the "S" terminal. The relay coil draws MA of current Vs a solenoid coil which it draws a huge amount of current.