I have a customer who has loop start lines coming into a MICS and being answered by a NAM Auto Attendant. They are having an issue when an outside call gets answered by the Auto Attendant and the caller does not make a selection and hangs up. About 15 seconds after the outside caller hangs up the receptionist phone begins to ring. When she answers she can hear full conversations of calls completely irrelevant to their business. The outside callers cannot hear her when she attempts to speak to them.
I have Trunk Supervision set up for the Trunks in the MICS. Is there a setting in the NAM that can disconnect the call when an outside caller hangs up while the main AA greeting is playing?
Any input would be appreciated.

Usually when something like that is happening with the auto attendant, it is because the LEC is not sending a proper disconnect condition (a momentary loss of voltage) when the originating caller hangs up.

Check to see if the LEC is sending such a signal. Also having the LEC test each line might be worth the while.

Mike Sandman sell a CPC tester (Calling Party Disconnect).

I used one to prove to Verizon that they were not sending the signal.