Legislation prohibiting smoking in public places in Wales came into force in 2007 and Ms Hart told AMs she was thinking about amending the rules to extend that ban.
She said: "I am actually considering an amendment to the smoke-free premises (regulations)... but we haven't come to any firm conclusions on this... some of the LHBs (Local Health Boards) are already looking at some of these issues."
Of the smoking she had witnessed at one hospital, she said: "I won't name the maternity unit, but when I go through the smoke to get into it... you start to think what on earth's going on?
"You know, we've got all these rules and regulations in place," she said.
But she said there was a need "to be realistic" about how difficult it might be to extend the smoking ban to all hospital grounds because some people would refuse to abide by it.
"Some people will still say, effectively, 'up yours, I'm going to smoke here'... we also recognise that when there's abuse it's poor staff then that try to rectify it [and] get an absolute mouthful," Ms Hart said.
The British Medical Association has called for smoking to be outlawed in hospital grounds, arguing that it was absurd to impose a ban on pubs and restaurants, but not on places where people go to be cured of illnesses.
But Simon Clark, director of pro-smoking group Forest, said a ban in hospital grounds would be "petty, vindictive and totally unnecessary".
He said there were "really horrific stories" of patients being "wheeled out in wheelchairs attached to drips and they're pushed a long way away from the hospital building".
"Hospitals are meant to be caring institutions and they need to live up to that and not discriminate against people who smoke a perfectly legal product.
"Nobody can say that anybody is at risk from somebody smoking in the open air," Mr Clark added.
Last month MPs supported a backbench amendment to the UK government's Health Bill to outlaw cigarette vending machines in England, Wales and Northern Ireland.
The Scottish Government has also decided to ban the machines.
Ms Hart said she was "quite supportive" of such moves.
"My officials are doing further work with Department of Health officials and I don't have any further [details] than that in terms of where we are... it's just in terms of quite a low-level discussion at the moment," she said.
"But I think that most people would agree that banning tobacco vending is the right way forward.
"We've done such a good job on the issues around tobacco advertising and smoke-free premises and everything, this is just another natural step, I think," Ms Hart added.
However, Mr Clark said there was "no justification whatsoever" for such a plan.
"It's claimed that this will help reduce youth smoking rates but the number of under-age smokers who use vending machines is absolutely minute, largely because it is the most expensive way to buy cigarettes," he insisted.
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