It's ok nab, sod them!
You can ask for a different doctor if you feel yours doesn't really understand. I don't think anyone can say that you won't attempt it again - how the hell do they know?
You can refer yourself to a CPN. Ring up your local hospital and ask for the Mental Health Unit, explain that you are currently under GP care but have attempted suicide once and you feel it is important that you have help asap as you are afraid that the suicidal feelings have not gone away.
These people will also help you to get the best care and are there to listen should you ever feel the need to just talk.
If you are anything like me, you'll feel worse at night. The Samaritans are very experienced at listening to and talking to people feeling suicidal. You might not think that just a simple act such as talking, would make that much difference, but it does.
Don't be fobbed off by people who have no idea what you are going through. Only you can say how you feel, only you know how potentially serious it is. Don't be afraid to tell them everything, and I mean everything, because I've a feeling you are holding stuff back. You're not telling them all you should for fear of being judged or condemned. Trust me, they've heard it before, possibly a million times worse. If you have the guts to speak out, you'll get their respect, because they know how hard it is to admit to those feelings. My GP once said to me that she knew if someone was covering up and that was the most frustrating thing. She knew they desperately wanted to tell her everything, but something held them back, and without that information, she couldn't help them. You can't refer someone on a 'feeling' that something is wrong.
Be brave.
The worst thing I ever had to do was to tell someone, when I was 8 months pregnant, that I hated the baby, I thought I might harm it and in fact, I thought I may be possessed. I was expecting judgement, condemnation, shock. But I got none of them, instead I was met with understanding and concern.