Without using the Time Hierarchy Theorem, is there any other way to prove P /= EXP
There is a way, actually! Kind of.

Consider two cases: 


$P \not= NP$. This case is easy: then $P \not= EXP$, because $NP \subset EXP$.
$P = NP$. It is known that $P = NP$ implies that there is a language from $EXP$ that has circuit complexity of at least $\frac{2^n}{10n}$, so $EXP$ is not a subset of $P/\mathrm{poly}$ in this case, let alone $P$. I will recall the proof here with more details if it is necessary, but that question has already been answered, for example, here: https://mathoverflow.net/questions/57828/p-np-exp-has-circuit-size-o2n-n (the answer is pretty sketchy, though, so feel free to ask for more details).