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Oct 10, 2017 at 23:06 comment added Hacon It seems that Prop 5.1 of arxiv.org/pdf/0902.0648.pdf by Koll\'ar-Kov\'acs gives a precise answer to this question in the LC case (after running an mmp you get a Mori fiber space $X'->S$ with general fiber $\mathbb P ^1$ such that there are 2 sections $E_1,E_2$ corresponding to the non-klt locus of $(X',D')$; moreover $(X',D')$ is plt. If you are indeed interested in the non-lc case, one can start by taking a dlt model and then running an mmp as in the proof of Prop 5.1 of arxiv.org/pdf/0902.0648.pdf. Does this help?
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