Justification of log f in DTIME hierarchy theorem
For n-tape TMs a tight time hierarchy result similar to the space hierarchy theorem is proven by Furer in 1982. The $\lg$ factor is not needed. 

The $\lg$ factor for the time hierarchy theorem stated in your post is only for single-tape TMs. Unless you are very committed to the single-tape model for some reason there is not a difference between space and time regarding the hierarchy theorems. 

There are some reasons and arguments for using single-tape TMs for defining time complexity classes, but the usage of single-tape TMs for defining complexity classes is not universal, e.g. see Lance Fortnow and Rahul Santhanam's [2007] where they use multiple-tape TMs.

The original reference for the time hierarchy theorem is Hennie and Stearns [1966]. They prove the theorem for two-tape machines. Odifreddi's Classical Recursion Theory cites them and Hartmanis [1968] and describes a proof that looks similar to the one in Sipser's book.

However the proof for single tape TMs in Hartmanis's paper does not use simulation simply. It distinguished between two cases: 1. running-time being $\Omega(n^2)$ and 2. running-time being $o(n^2)$. 


For the first case it uses a simulation, and it seems that one can get rid of the $\lg$ factor if the simulations can be done more efficiently. 
For the second case, the paper directly gives a language for the separation and doesn't use simulation at all. This uses particular properties of single-tape TMs with sub-quadratic running-time.


One should note that single-tape TMs with time $o(n^2)$ are not as robust and there are quadratic lowerbounds (e.g. for Palindroms) on single-tape TMs whereas a two-tape TM can solve such problems in linear time. 

As I said above, unless you are committed to single-tape TM model for some reason, even when time is sub-quadratic, there is not a gap to fill, the time hierarchy theorem is as tight as possible.

PS: if we are using multiple-tape TMs, i.e. a Turing machine in the class can have fixed but arbitrary number of tapes Fürer's result does not apply.


Martin Fürer, "The Tight Deterministic Time Hierarchy", 1982
Piergiorgio Odifreddi, "Classical Recursion Theory", vol. II, 1999
(page 84)
Juris Hartmanis, "Computational Complexity of One-Tape Turing Machine Computations", 1968
F.C. Hennie and R.E. Stearns, "Two-Tape Simulation of Multitape Turing Machines", 1966
Lance Fortnow and Rahul Santhanam's paper "Time Hierarchies: A Survey", 2007