Is there any way an incoming called can leave a voicemail directly, without waiting for the call to ring?

When someone from outside our organization calls in, is there a way for the caller to send their own call straight to voicemail? We have an answering service that helps us answer calls and we want them to try to transfer calls to us when possible, but there are times when it would be better for them to just send the call straight to our voicemail.