Schramm-Loewner evolution (SLE) is a random process that gives a useful description of fractal curves. After its introduction, many works concerning the connection between SLE and conformal field theory (CFT) have been carried out. In this paper, we develop a new method of coupling SLE with a Wess-Zumino-Witten (WZW) model for $SU(2)$, an example of CFT, relying on a coset construction of Virasoro minimal models. Generalizations of SLE that correspond to WZW models were proposed by previous works [Bettelheim {\it et al.}, Phys. Rev. Lett. {\bf 95}, 251601 (2005)] and [Alekseev {\it et al.}, Lett. Math. Phys. {\bf 97}, 243-261 (2011)], in which the parameters in the generalized SLE for $SU(2)$ were related to the level of the corresponding $SU(2)$-WZW model. The present work unveils the mechanism of how the parameters were chosen, and gives a simpler proof of the result in these previous works, shedding light on a new perspective of SLE/WZW coupling.