I think you're right that it needs to be more general support, someone I loved had just the last 7% of her life 'covered' by suicide hotlines and 26 years of suffering before them. The final days, weeks, months, it was all just a delaying measure at that point because she'd already come to terms with what she wanted to do.
I don't really think the solution is more hotlines, even generalized ones because you have to want to get help every single time. I think the solution is to shine so much light on mental illness it loses all of the mystery and stigma and becomes as widely discussed and debated as the following 'mentally healthy' ways to kill yourself or someone else:
- Euthanasia, or "mercy" killing because we are glad someone's suffering has ended
- Abortion, literally deciding to end another person's life who is incapable of asking you not to
And of course, the preferred method of suicide that most people will use:
- a slow poison of shitty diets and deadly vices
We never even discussed mental illness or even suicide in high school like we did ethical dilemmas.