Making guns less available does not reduce suicide but merely causes the person seeking death to use another means.
While gun-related suicides were reduced by Canada's gun control legislation of 1978, the overall suicide rate did not go down at all: the gun-related suicides were replaced 100% by an increase in other types of suicide -- mostly jumping off bridges.
"The authors describe suicide rates in Toronto and Ontario and methods used for suicide in Toronto for 5 years before and after enactment of Canadian gun control legislation in 1978. They also present data from San Diego, Calif., where state laws attempt to limit access to guns by certain psychiatric patients. Both sets of data indicate that gun control legislation may have led to decreased use of guns by suicidal men, but the difference was apparently offset by an increase in suicide by leaping. In the case of men using guns for suicide, these data support a hypothesis of substitution of suicide method."
Source: "Guns and suicide: possible effects of some specific legislation," Rich, Young, Fowler, Wagner, and Black, The American Journal of Psychiatry March, 1990