rythmic squeak?

My '06 corolla is kept in the garage. When I start it there is a rythmic squeak. It sounds like a pulley squeaking but my long time auto shop (not Toyota dealer) has installed a new belt and checked every pulley with a stethoscope and can't find it.

The squeak sometimes goes away but returns the next morning. I assured there was no coolant or oil dripping on the belt or pulleys.
I'm afraid to take it to Toyota so they can charge me $200-$300 for something I don't need.

I was hoping someone on this forum has experienced the same thing and corrected it.
Thanks in advance.
The mechanic just called and said they put another new belt on (for free).
A new one was put on recently and it started squeaking again so we'll see if this corrects the rhythmic squeaking that goes away and then comes back when the engine is cold.