Assuming Twilio allows multiple calls to the same number to happen simultaneously (I've never tested this), you could build a lookup table in your app that stored active calls, then give the user a control to swap between active calls. You could send a push notification to notify the user of additional incoming calls (a vibration) if there's not a way to play a sound or otherwise notify the user through the active Twilio call. (There should be a way to send an audio notification, though.)

This is all untested, so I can't speak to whether or not Twilio can/will support that setup, but — assuming multiple calls are allowed through their service — it should be possible using something like that.

Good luck!