Came up with a mod to further randomize it, relies on seconds.
Next Rolled Number: Direction of Second Hand.
North: 56-04 Seconds
North-East: 5-10 Seconds
East 11-19 Seconds
South-East: 20-25 Seconds
South: 26-34 Seconds
South-West: 35-40 Seconds
West: 41-49 Seconds
North-West: 50-55 Seconds