Publishing a second paper which is an extension of the first paper- same methodology, problem formulation and literature review
If the first paper has not been published yet, you may want to contact the editor who is handling it and ask him whether it would be possible to withhold publication (in case it gets accepted) and publish the two papers together as a series of papers. In this case setting up the context for the second paper should be fairly easy, as you wouldn't need to rewrite the introduction and literature review from scratch.

If that's not possible then you'll have to rewrite introduction and conclusions so that the paper can stand on its own. Of course, you can be a bit more concise and write upfront that the second paper is an application (or an extension) of the first one, and that additional details can be found in the latter. However, note that if you rely too much on another paper some journals (depending on the field, I guess) might complain about a "lack of novelty".