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Use a section of rubber hose as a stethoscope to attempt to narrow down the noise area.

If a rod bearing the noise would get louder as engine load increased.

Sometimes the A/C pulley can start to rattle when the compressor is not engaged. Try listening with AC compressor on and off.

There is not much in the back of the engine.

Might be valve lifters. This has a more rapid ticking pace the a bearing would.

Can't tell if about exhaust leak. The easiest way to find this is to get under the car and move your hand around pipes. If a leak you will feel it. You don't have to touch the pipe to do this.

Fuel injectors can make noise.

Might post at Toyotanation.

 

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Sounds like either valve train or fuel injector noise. Fuel injector noise can be kind of loud but normal.

The fact the noise goes away somewhat as the engine warms up may mean the valve train clearance is decreasing.

Perhaps the replacement exhaust system was not as thick walled allow more noise to come through. Just a guess.

 

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