Recently, it was shown that modified mimetic gravity, with a f(□ϕ) term, results in a singularity-free model of gravity, for both cosmological and black hole spacetimes [1,2]. In this work, we analyze this model further and show that, since the function f was tuned to vanish rapidly for small values of the argument, the non-singular bounce relies heavily on a subtle branch changing mechanism for the multi-valued function f. Furthermore, this mechanism has interesting implications for the proposed link between this model and loop quantum cosmology.