1976 Toyota Corolla Starting Car After Been Running While Example

If you're hearing a single rather loud clunk from the starter each time you turn the ignition switch to "crank", that is typical of worn contacts inside the starter solenoid. Most people just replace the entire starter, but the design looks identical to the Nippendenso starters used on Toyotas and Chryslers in the '90s. For those you can find repair kits in hardware stores and farm and home supply stores. They come with four contacts. You use the two that match those in your solenoid now. The secret is not in having to wait an hour for it to crank. It will crank eventually when you cycle the ignition switch enough times. The number of times you have to do that will increase as the contacts continue to wear away.

Unfortunately, I do not hear any sound (such as the loud clunk) from the engine or the starter. When you turn the key, I hear just a click from the relay mounted on the inside on the front passenger side and the fuel pump humm....absolutely nothing to indicate that the starter is starting to crank

Measure the voltage on the smaller wire on the starter. You should find 12 volts when a helper turns the ignition switch to "crank". If you do not find 12 volts, the relay has a bad contact or there's a break in the wire feeding it. Look at the color of that smaller starter wire, then find that same color wire at the relay and measure the voltage there.