Using StiffnessSwitching seems to have smoothened out the solution. Thank you for the suggestions.
tA = NDSolve[{Subscript[PDE, in], Subscript[PDE, out], IC,
Subscript[T, f, o][\[Tau], 0] == 10,
Subscript[T, f, in][\[Tau], L] ==
Subscript[T, f, o][\[Tau], L]}, {Subscript[T, f, in], Subscript[T,
f, o]}, {\[Tau], 0, 3600}, {z, 0, L},
Method -> "StiffnessSwitching"]