I have thought about this question before, but at the moment I can't remember links or references. Nonetheless, many years ago I thought of a sketch of an argument that should eventually work to prove that it's always possible. Namely, start with any smooth embedding of $S$ into $\mathbb{R}^3$. This surface has some conformal structure, and any other conformal structure is described by an ellipse field well-defined up to a scale. Then it is intuitive that that you can match the ellipse field by "wrinkling" $S$, in other words by making $S$ locally wavy in the long direction of the ellipses. More rigorously, $S$ is replaced by a graph of a function that is locally sinusoidal. You don't even have to match it exactly; approximations are good enough if you can encircle the desired conformal structure in Teichmuller space. There is a similar trick for a different purpose in the video "Outside In".
Of course it's messy to make all of the required transitions between the patches of wrinkles on the surface. Moreover the surface is not flat, only flat to first order, and a function cannot be graphed rectilinearly, only approximately so. I think that Garsia's paper uses the same idea or a similar idea, but he slogs through all of the approximations to make it work.