I call my concierge's number and it says she isn't available press a number to be connected to the next one. This works really well as I get someone when I call and have only had to wait a min or two. Their number should be on the intro email they sent you introducing themselves.
I emailed her once and never received a response it was on a GOH reservation so I wasn't worried about the cost going up. Calling your concierge is the way to go, as I can understand them being busy, but putting you through to the next one works out great. This way it seems like a benefit that works as opposed to one that doesn't work.