Coil whine is described as a high frequency vibration within the coil. Certain load situations can increase/decrease those vibrations - In your case, i'm under the impression it's being caused by one of your lessor voltage rails (such as 5v used for stand by). Once the rail is loaded up, the vibration frequency changes, rendering the noise inaudible.
Doesn't help the situation I know but there is not much (if anything) outside of replacement you can do to get rid of it. If it was occuring while you were under load, you could change the load to possibly get rid of the whine but it whining on standby doesn't leave anything to change.
I don't see how it could be caused by the SSD, unless it is also getting standby power. I suppose you could try unplugging the sata power to the SSD and seeing if the noise goes away (that would suprise me).