How can I change the ringing noise when I place a call? I'm not asking to change the ringtone for contacts (what plays when they call me). Now am I asking how to change the outgoing ringtone--so that when people call me they hear a custom ringtone while waiting for me to pickup.

I just want to change the sound when I place a call and it rings while waiting for the other party to pickup. I'm using CyanogenMod 7 on a rooted T-Mobile MyTouch 4G. The reason I want to do this is that the ringing noise is quite loud. If I have the Phone Call Volume all the way up, so I can clearly hear people on the line, then the ringing is loud enough to be painful. If I have it turned down so the ringing noise isn't overpowering, I can't hear people on the line after they pick up--and don't want to have to turn the volume up and down every time I place a call.

(I did consider using Tasker to change the Phone Call volume when ringing and when the other person picks up, but I didn't see the right states/events to be able to do that).

Is there a way to change what ringtone is used, so I could just pick a quieter tone?
Your question is a touch confusing, especially due to the use of the word 'ringtone', but I think this is the crux of what you are asking: 


  I just want to change the sound when I place a call and it rings while waiting for the other party to pickup.


The answer is no.  That sound is generated by the communications network, and your phone is just playing it to you.  Your phone has no native way to differentiate between that sound and the sound of the person on the other end (or their voicemail).

In theory, an application could be written that would listen to all the sound during a call and adjust volumes depending on what it hears.  I performed a fairly exhaustive search, and found no applications that currently do this.