Staff have been stationed there to prevent overweight vehicles crossing while repair work continues.
Workers were withdrawn from the Menai Bridge over safety concerns earlier this week following a confrontation.
Deputy transport minister tells Senedd "detailed options" for third crossing to be looked at.
The Menai Suspension Bridge, in Anglesey, is closed for the third time in about 24 hours.
BBC presenter Teleri Glyn Jones heard the call for change as she toured Wales ahead of the election.
The plans are set to meet a "significant and growing demand" for accommodation for older people.
It shut in 2022 because of defects in the hangers, which link suspension chains and the road.
It will be checked every Wednesday when traffic will be managed with signs.
The bridge unexpectedly closed last Saturday after engineers discovered that repair work was needed.
