The paper deals with a quantitative aspect of the well-known Enflo-Pisier theorem on the existence of uniformly smooth renormings of superreflexive (in particular, uniformly convex and uniformly non-square) Banach spaces. A typical result: Let the modulus of continuity of a Banach space $X$ with a local unconditional structure satisfy the inequality $\delta_X(\varepsilon)\geqslant c\cdot\varepsilon P$. Then $X$ admits an equivalent $q$-smooth renorming for any $q$ satisfying $$ q<\log2/\log[2(1-c\cdot2^{-p/2})]. $$