STARTER ISSUES

Inside the starter relay (solenoid) the brass contacts that bridge the electrical current are probably out of kilter. "New" just means no one owned it before you, doesn't necessarily mean good. The guy in the factory could have put it together on his Friday.

Did you check volts on the small wire going to the relay from the ignition switch? If it is getting power with the key off than you have a bad ignition switch (in the steering column). Good luck.