Sounds a bit like a solenoid problem to me. If this "new" starter is a
re-built. it is most likely that nothing was done to the solenoid if it
worked at the time. You can remove starter, open it up, and clean up the
solenoid or replace it with a generic solenoid or, if its still under
warranty, hand it back to whomever you purchased it from.
The above was for the new, post-68 starter. the solenoid is a separate item
mounted on the passenger side wheel well in the older models.
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