Jet engines have 3 BASIC limits - Pressure limit (internal pressures), Temperature limits (Turbine temperatures), and Speed limits (N1, N2, N3 etc.).
At low temperatures (and therefore high air density), it's possible to reach the internal pressure limit well before the Temperature or Speed limit. (This will show as a constant EPR on EPR gauged aircraft, or steadily increasing N1 with increasing temperature on N1 gauged aircraft).
As Air Temperature increases, more and more fuel must be injected, with rising Temperature and Speed, until the limiting turbine temperature is reached), so far, thrust has been constant, hence the 'Flat' rating.
As air temperature increases further, fuel must be limited to maintain turbine temperatures at their limits (Both EPR and N1 now begin to decrease). The engine has now reached full rating, and thrust continues to decline as air temperature begins to rise.
At much higher altitudes (usually), the engine speed limit is reached. This is not usually a problem at typical takeoff altitudes.
So....Flexing to a temperature below the flat rating limit temperature would achieve nothing, except full thrust.