How old is this HDD? While they're not silent, they shouldn't be overly loud to the point of being an annoyance....
They're are "anti-vibration" mounts you can purchase, so creating your own shouldn't be a problem. Full wrapping it though, could potentially result in temperature issues....although I wouldn't expect too much of a concern. Where it could be problematic, would be limiting the ability to equalize air pressure.
The "do not cover this hole" warnings you see on HDDs are for a purpose. The hole (with a filter inside) allows the drive to equalize air pressure to account for different pressures/altitudes between location of manufacture and location of use, without this, the drive can pretty easily face catastrophic failure.
For example, a drive manufactured in Taipei would be ~1,100 Ft altitude. Those drives could be utilized at various altitudes. An example of those extremes could be NYC @ ~30Ft to Mexico City @ 7,500 Ft. Quite the variety there.