Maritime and Coastguard Agency say man taken to hospital via helicopter following incident on Saturday night.
Operators have said they are now confident the risk of ice and snow falling from the cables has passed.
It comes as a yellow warning for snow and ice remains in place across much of Scotland.
Drivers warned of delays on main roads with up to 30cm of snow due over the next few hours.
The crossing connecting Edinburgh and Fife is closed to all traffic until further notice.