Sounds like to me that the voltage drops off at the solenoid coil causing it to pull in with less force than is needed to make good contact to keep from burning up the contacts in the starter.

Makes sense to me, but you should first try jumping from the main battery terminal on the starter to the solenoid terminal to see if it makes any difference before doing any mods.

If that helps go ahead and use a secondary relay to get good voltage at the coil.