If the starter solenoid "clunks" when the key is turned to Start, but the starter motor doesn't start, but the start motor will turn over the engine when connected directly back to the battery positive post, then it's most likely that the solenoid switch contacts are not being connected properly.Starter will work and start the car if by pass ground and go straight from hot side of battery to starter. Any suggestions short of installing a momentray switch?
It's also possible that the positive cable from the battery positive post to the solenoid is bad -- sometimes it's not the clamps but where the cable copper wire is crimped into the end clamps. Corrosion between the copper wire and the end connectors can prevent sufficient current from flowing. Same for the ground side.
If there's enough access, measure the voltage at the stud where the battery positive cable connects to the starter solenoid. With the key off, it should be the same as the battery voltage. Then turn the key to Start. The voltage will drop to 11 V or less if the starter motor is being connected; if it drops only a small bit, then the starter motor is not being connected through the solenoid contacts. The "new" starter could be defective.