You can't jump from that point straight to God, not from a position of pure mathimatical logic anyway.In Descartes' mind, the jump from his own existence to the existence of God was possible because the ontological argument for God (the Fifth Meditation) is, as its name implies, based on existence. Descartes claims he can prove the existence of God merely from the idea of God. And that's what the ontological argument purports to do.
What makes Descartes' argument peculiar is that it takes as a premise the idea of a perfect being. Descartes' cogito argument leads to the conclusion that all of the ideas that we find in our minds exist as ideas in our mind, and the evil demon can't deceive us about that. Any idea that I find in my mind, I know to exist as an idea in my mind. So if I have the idea of a perfect being, I know that that exists as an idea.
The question is, does the idea of a perfect being correspond to anything outside me? For example, using my idea of a desk, I can ask whether the idea of a desk corresponds to anything outside me. There seems to be a desk; does that 'seeming' to be a desk correspond to anything real outside me? And you can ask the same question about the idea of perfect being. Does that correspond to a real perfect being outside me?
Descartes argues that all we have to do is to think of this idea of a perfect thing, to analyze this idea, and we'll come to see that such a being necessarily exists. That is, he argues from the idea of such a being to the necessary existence of such a being.