In their seminal paper, Jeff Cheeger, Mikhail Gromov, and Michael Taylor derivated bounds on the injectivity radius of Riemannian manifolds with bounded sectional curvature of the form: $ inj(p)\geq r \frac{Vol(B_M(p,r))}{Vol(B_M(p,r)) + Vol(B_{T_p(M)}(0,2r))} . $ Their paper assumes that the involved manifolds are of $C^{\infty}$-type.

Since then, has anyone derived refinements/extensions for manifolds of lower regularity; i.e.: $C^k$ for some $1\leq k<\infty$?