I think it's always good to keep in mind that these "accessible" microstates and their "increase" or "decrease" are not real physical properties or processes of the system. Strictly speaking the system has only one accessible microstate at time $t+\mathrm{d}t$: namely the one determined by its microstate at time $t$ and the equations of motion.

The accessible microstates are those that you consider most likely candidates to be the actual microstate, when you don't know the latter. This set is determined by the macroscopic information you have – in this case the total energy, volume, and number of molecules. If you have different information, their number is different. If you have full information, their number is 1.

It may happen that during that brief contact energy passes from the system with lower mean kinetic energy to the one with higher.

Coming to your question, the change in number of likely microstates with respect to a small change in the mean energy must be computed with respect to the mean energy of the system. That is, for the sistem with higher mean energy, say $E_+$, the change is $$-Q\;\frac{\partial \varOmega(E)}{\partial E}\Biggl\rvert_{E=E_+} \;,$$ and for the other, with lower mean energy $E_-$, it's $$+Q\;\frac{\partial \varOmega(E)}{\partial E}\Biggl\rvert_{E=E_-} \;,$$ where $Q>0$ is the small amount of energy exchange, the same for the two systems apart from its sign.

If you multiply these changes you'll see that the change in total number of likely microstates is positive from the fact that $E_+ > E_-$.

I think it's always good to keep in mind that these "accessible" microstates and their "increase" or "decrease" are not real physical properties or processes of the system. Strictly speaking the system has only one accessible microstate at time $t+\mathrm{d}t$: namely the one determined by its microstate at time $t$ and the equations of motion.

The accessible microstates are those that you consider most likely candidates to be the actual microstate, when you don't know the latter. This set is determined by the macroscopic information you have – in this case the total energy, volume, and number of molecules. If you have different information, their number is different. If you have full information, their number is 1.

It may happen that during that brief contact energy passes from the system with lower mean kinetic energy to the one with higher.

Coming to your question, the total number of likely microstates before contact is $\varOmega(E_+)\times \varOmega(E_-)$, where $E_+$ is the energy of one system and $E_-$ of the other.

The change in this number for a small exchange of energy is $$-Q\;\frac{\partial \varOmega(E)}{\partial E}\Biggl\rvert_{E=E_+} \times \varOmega(E_-) + Q\;\varOmega(E_+) \times \frac{\partial \varOmega(E)}{\partial E}\Biggl\rvert_{E=E_-} \;,$$ where $Q>0$ is the small amount of energy exchange, the same for the two systems apart from its sign.

If compute this you'll see that the change in total number of likely microstates is positive from the fact that $E_+ > E_-$.