More efficient Cook-Levin reduction
The obliviousness of the Turing machine is very crucial for Cook-Levin construction to run in time $t(n) \log t(n)$. If the Turing machine is not-oblivious (that is the head position at $i^{th}$ step of computation is not a function of input length ) then, on different inputs, the head position at $i^{th}$ step can be at various positions. But then writing a constant size formula which depends only on the two cells adjacent to head position and head position itself, to ensure correctness of transition from one configuration to the next, cannot be written without knowing the input(As Kaveh pointed out, it is easy to get a $t(n)^2$ formula without the TM being oblivious). But whereas if the Turing machine is oblivious, we can know the position of head at step $i$ on a input of length $n$. Hence can write this formula for correctness between two adjacent configurations. So I guess you should be asking for a better simulation of oblivious Turing machines which beats $O(m \log m)$. It is already asked here : Oblivious Turing Machine emulation lowerbound.