Over half of patients and staff in Swansea want a total ban on smoking in the city's hospital grounds, according to a survey.
The health trust's survey will run until the end of March
Swansea NHS Trust has so far received 1,000 people's views on how wide-ranging the ban should be.
It already stops people from smoking indoors in line with new legalisation that takes effect in Wales on 2 April.
Other hospital trusts in Wales who already operate smoking bans in grounds admit they can be hard to enforce.
Around 1,000 people have responded to the poll in Swansea, which will run until the end of the month.
The Swansea trust has put forward three options.
The first sticks to the legal minimum which would allow people to smoke under canopies, outside entrances and elsewhere on trust grounds.
But this has only attracted 10% support.
Another 39% said they wanted to ban smoking from near buildings but allow it further away.
But 51% want to see a blanket ban.
A trust spokeswoman said although the results would not be binding the aim was to gauge responses for future decisions.
"The trust is currently looking at its smoking policy to see if it will go further than the new legislation in the months which follow," she said.
Under Welsh Assembly Government legislation smoking will no longer be allowed in enclosed public places from 2 April
Pat Dwan, Unison health spokesman for Swansea, who represents many hospital staff, said the union was in favour of the new legislation.
But he said it was difficult to see how wider ranging bans could be enforced.
"What we don't want to do is drive it underground and go back to the bad old days where a nurse would nip into the linen cupboard for a quick smoke," he said.
"There has to be somewhere for these people to go to wean themselves off."
Carmarthenshire NHS Trust has already announced its smoking ban will include hospital grounds and car parks.
From 14 March, which is National No Smoking Day, staff and patients will no longer be allowed to smoke in any part of its hospitals including entrances, grounds, car parks and staff residences.
The trust's workplace health co-ordinator John Price said other UK hospitals had already introduced similar restrictions.
"Even though some people will disagree about the stance taken, we have an overwhelming responsibility toward improving health and reducing smoking must be a key priority," he said.
Cardiff and Vale NHS Trust's smoking ban has applied to hospital grounds for the last two years.
It hopes the introduction of the new law will make this easier to enforce.
Shelters are being put up to encourage smokers who insist on lighting up in hospital grounds to move away from the entrances.
Previously, patients who had received bad news were sometimes allowed to smoke inside but this will no longer be the case. Smoking rooms at two hospitals will also be removed.
The only exception under the new law will be at mental health institutions, including Whitchurch hospital and the Llanfair unit at Llandough.
A ban including hospital grounds is also already in place at North East Wales NHS Trust hospitals and clinics, although enforcing this outside the buildings can be difficult.
The new law will be a chance to re-promote this ban said the trust, adding that it was particularly keen to prevent smoking at entrances.
At the Royal Gwent Hospital in Newport, a trust spokesman said it was "likely" that the ban would extend to the grounds eventually.