Given a minimal Cantor system $(X,T)$, a topological speedup of $(X,T)$ is a dynamical system $(X,S)$ where $S$ is a homeomorphism such that $S(x)=T^{p(x)}(x)$ for some function $p:X\to \mathbb{N}$. We assume the function $p$ is continuous (and thus bounded) and the resulting system $(X,S)$ is minimal. One can ask what properties of the underlying initial system $(X,T)$ are preserved under minimal bounded speedups. We investigate the class of Toeplitz flows, which are minimal symbolic almost one-to-one extensions of odometers. Although the minimal bounded speedup of an odometer is always a conjugate odometer, we demonstrate that the minimal bounded speedup of a Toeplitz flow need not be Toeplitz. We then provide sufficient conditions to guarantee that the minimal bounded speedup will be a Toeplitz flow; in this case, it is never conjugate to the original Toeplitz flow but has the same underlying odometer.