Can we get a better time hierarchy theorem?
IMHO, there cannot be better, stricter Time Hierarchy Theorem everywhere.

To improve this theorem, you need faster Universal Turing Machine. The $log(n)$ overhead of the currently fastest simulation technique seems to be unavoidable, especially for large enough time bounds, since with more time resource the simulated TM can do crazy things that the UTM cannot assume some regular property to do the simulation faster.

With small time bounds, there may be some improvements. But, its consequences are negligible.