First of all, it's noteworthy Australia can be an example for at least some of the US, especially since the bulk of our suicides are done with and facilitated by guns. The suicide rate of Australia in 2014 was 12.71 per capita and is worrying but the gun suicide rate is under 1 per capita.
However, Australia presents a second harsh lesson with suicide. Low suicide towns in Australia have seen "recent improvements in the availability and geographical access to focused mental health initiatives" that "were associated with a reduction in the historically high rates of both youth and farmer suicides" while "In contrast HS towns, publicly funded health services were described as being very difficult to access, poorly advertised, or as having prohibitive delays for acute care."
It's noteworthy that suicide is extremely correlated with systemic inequality like poverty, as well as gender and drug abuse but it is also correlated with age, and Australia is an aging country with worrying health indicators even with a strong health care system.
To what degree these issues are involved needs further analysis than a trend line. Australia did in fact introduce more access to health care in 1984 which, when put into policy here, does reduce suicides. But in 1996 it shifted to a private/public system, seemingly leaving behind those using the public one.
All of this is important -- for Australia. Consider however that the United States citizenry are at present not experiencing a hanging epidemic but a gun suicide one (and are hopefully moving towards greater access to health care, we'll see). Looking at overall rates of gun violence in Australia is not a bad place to start. So what did happen to the gun violence rates after 1995 in Australia?
According to only just-dated but possibly relevant research by the Medical Journal of Australia (1988-2007):
Considered nationally, the falls in male suicide were due to significant reductions in shooting, gassing and poisoning, and occurred despite an increase in suicide by hanging. There were similar changes in the methods of suicide used by females. These results are consistent with the hypothesis that that measures to control the availability of firearms, the requirement for new cars to be fitted with catalytic converters and the decline in the prescription of tricyclic antidepressants have resulted in a decline in total suicide rates.
Here is the per capita gun suicide rate following 1995's gun buyback program: