The reason you hear it most often after cold starts is as soon as the engine is running, the ECM retards the spark, to heat the exhaust quickly so the cats light off. When the spark is that retarded, some combustion is going on downstream of the exhaust valve and that makes the engine more noisy. Once sufficient time has past, the timing advances and the noise is reduced or goes away.
If the problem is a small exhaust leak, the leak will be louder after start up when spark is retarded.
On some engines, GM surrounds the exhaust head pipes, just downstream of the joint between the pipe and the manifold, with a tight fitting, waffle looking shield. I have also found that if that shielding is not fully closed at its seam, the gap can allow cold start-up up noise to escape and the sound is very similar to an actual leak but...there is no leak.
You have to get under the car while its running and use a good automotive stethoscope to fully diagnose this problem.