The starter solenoid clicks when it either can't get enough voltage/power/amperes from the battery, or the starter motor and its integrated starter solenoid itself is bad.
The first cause is usually fixed with a new battery. Since you have a new battery, verify both battery connections are clean (!) and TIGHT. They may look fine but if you haven't wire brushed the inside of the battery connections to assure it has a good solid electrical connection to that new battery, that could be your cause.
If the connections are ok and TIGHT, then it's like just a bad starter motor. If I needed a new starter motor, I'd go right to the dealer as the OE starter motor is about the best quality there is. The cheapy "lifetime warranty" starter motors sold by the chain stores like Autozone come in dead-last on the list of reliable high-quality starter motors.