1990 Toyota Corolla starter wouldn't engage

The starter works fine now, but recently when the temperature dipped to 0 F, it took several turns of the ignition key--when the solenoid was heard to click in with each turn--to get the starter to engage and start the engine. After the first start that morning it was fine.

I cleaned the battery terminals that day thinking that was the problem. But the next morning with the same temperature the same thing happened. This time I kept the solenoid engaged and after a second or two the starter engaged and started the engine. Almost as though ice had formed on the contacts to the starter that the current had to melt before metal to metal contact could be had between the starter and battery cable.

Since then 0F temperatures haven't returned and the car has started without fail.

Thanks for your reply.