It is a subset trivially by their definition, even stronger: $\mathsf{DTime(o(f))} \subseteq \mathsf{DTime(O(f))}$. This is because $o(f) \subseteq O(f)$.
However the properness of inclusion is not correct. In fact there are functions $f$ which the proper inclusion doesn't hold even for $o(f/\lg f)$. (The hierarchy theorem uses the fact that $f$ is time-constructible function, otherwise the theorem does not hold and they can be equal.)
Obtaining a stronger version of the hierarchy theorem (under the same assumptions for the the hierarchy theorem) is a long open problem in complexity theory. The diagonalization result separating the classes is based on the universal simulation results and the best known simulation results need that $\lg$ factor increase in time. AFAIK, it is not known if the simulation can be done more efficiently. If it can be done more efficiently then we can obtain stronger versions of hierarchy theorem.