without close analysis I make the following suggestion as to why there is an 'apparent' pickup in outgoing LWR, at least over the period your graph exhibits.
there are 3 modes of heat transfer; conduction, convection and radiation.
conductive heat transfer is a linear affair, so if the temp difference doubles then rate of heat transfer doubles.
convective ... don't worry, its complicated, e.g. Nu=Re.Pr, amongst others.
radiative heat transfer is non-linear (T4), so if temp of object doubles then radiation goes up x16.
So if Earth's average temp went from 16 to 17 C (289 to 290 K), a 0.346% increase in absolute temp, then the radiation goes up by 1.39% [100 x (2904 - 2894)/2894]
This could well explain the OLR radiation curve going from slightly decreasing to slightly increasing while there is only a long period linear increase in temp.
