Consider (a hyperbola after substitution) $\log(x)\log(y)=1$ revolved about the line $f(x)=x.$ Consider a projection from the origin which maps points on the revolution of $\log(x)\log(y)=1$ to the revolution of $g(x)=1-x$ about $f(x),$ bijectively. The projection is given by the revolution of $h_n(x)=x^n,$ $n\in \Bbb R$ about $f(x).$
What metric will make this a model of hyperbolic geometry?
Notes:
a) $\log(x)\log(y)=1$ is a hyperbola with a substitution, and so its revolution is a hyperboloid with a substitution.
b) The revolution of $h_n(x)$ starting from the origin, passes through the hyperboloid and then through the disk, so it projects points from the hyperboloid to the disk.
There's nothing special about your example. Any smooth 2-manifold $M$ diffeomorphic to the Poincaré disc $\mathbb D^2$ has a Riemannian metric making it isometric to the hyperbolic plane: simply choose a diffeomorphism $f : \mathbb D^2 \to M$; let $\mu$ denote the standard hyperbolic Riemannian metric on $\mathbb D^2$; and then put the metric $f_*(\mu)$ on $M$. Probably you can use the projection in your note (b) to construct $f$, if you first use a diffeomorphism from $\mathbb D^2$ to the disc mentioned in (b).