Have you tried looking into the theory of Schwarz-Christoffel mappings? These are conformal maps of the upper half plane to a polygon (bounded or unbounded), so you could approximate your region by a polygon, and then apply SC maps. I know that this is different from the original question, but depending on what application you have in mind, it might work as well.

SC mappings have many interesting properties. First of all, for simple geometries (say, a triangle mapping to a disc) they can be explicitly written down, and for more complex geometries they involve a simple contour integral ("simple" in the sense of being easy). Secondly, there is a beautiful book by Driscoll and Trefethen which has many important examples, and for those examples that aren't in the book there is a very fast Matlab package written by Driscoll. I'm an applied mathematician, but ever since learning about SC maps and the tools available to deal with them, I almost never have to resort to using other conformal maps.

Last, this might be somewhat conjectural, but if you're interested in complex approximation theory, you could also look into Faber polynomials and how they are defined in terms of conformal maps. Basically, the $m$th Faber polynomial $f_m(z)$ is the polynomial part of the $m$th power of some uniformizing map of your domain, and the interesting fact is that the level sets $|f_m(z)| = 1$ provide "optimal" approximations (in some sense) to the boundary of the original domain. This is also explained in the book by Driscoll and Trefethen.

Have you tried looking into the theory of Schwarz-Christoffel mappings? These are conformal maps of the upper half plane to a polygon (bounded or unbounded), so you could approximate your region by a polygon, and then apply SC maps. I know that this is different from the original question, but depending on what application you have in mind, it might work as well.

SC mappings have many interesting properties. First of all, for simple geometries (say, a triangle mapping to a disc) they can be explicitly written down, and for more complex geometries they involve a simple contour integral ("simple" in the sense of being easy). Secondly, there is a beautiful book by Driscoll and Trefethen which has many important examples, and for those examples that aren't in the book there is a very fast Matlab package written by Driscoll. I'm an applied mathematician, but ever since learning about SC maps and the tools available to deal with them, I almost never have to resort to using other conformal maps.

Last, this might be somewhat conjectural, but if you're interested in complex approximation theory, you could also look into Faber polynomials and how they are defined in terms of conformal maps. Basically, the $m$th Faber polynomial $f_m(z)$ is the polynomial part of the $m$th power of some uniformizing map of your domain, and the interesting fact is that the level sets $|f_m(z)| = 1$ provide "optimal" approximations (in some sense) to the boundary of the original domain. This is also explained in the book by Driscoll and Trefethen.