2001 Tacoma - Chirping

I have a 2001 Tacoma Prerunner. My truck makes a fast chirping sound when applying the break at a stop. It also will make a few quick chirps when going over speed bumps, potholes, etc. It will also very rarely chirp if the truck is in park.

The chirping sound can be stopped by pushing or holding the exhaust pipe. Once released though, the chirping starts. I have greased all the exhaust hangers thinking it may have been rubbing somewhere and vibrating and that did not help.

I cannot find the exact problem/cause of the chirp. Anyone have similar issues or ideas?
Thanks

 

Kill the bird

 

Could be your leaf springs. Toyota leaf springs always squeak. And I guess it might sound like a "chirp" to some.