Lower bounds and class separation
I will try to answer the question more generally so does not depend on the specific problem. Let $L$ be a language in $DTime(O(t(n))-DTime(o(t(n))$. Can we use padding to get a language in  $DTime(O(s(n))-DTime(o(s(n))$ where $s(n) \in \Omega(t(n))$ is an arbitrary (time constructible) function?

The answer is no:


  the complexity class separations using padding translate downwards but not upwards, see Why do equalities between complexity classes translate upwards and not downwards?.