I may have been guilty of fixing a symptom and not the fault. You probably don't need to tighten up the rim around the glass.
Although the noise had gone I noticed the mirror was still visually vibrating a little. So I pulled it off again, centred the spring, put a small felt pad under the spring and duct taped it in place. No noise or vibration now. This alone probably will cure the noise.
To get the mirror off I hold it in both hands, one each side, with palms on the mirror palms facing the screen. A sharpish push towards where the screen meets the dash is required to persuade the spring to allow the two parts to slide apart. You are effectively pushing down and away from you.
Do be careful, get it wrong and damage could be done to the screen.