It would make sense to carry the evaluation to a given position to the position one move before, then you have the move which happened in the game already analysed. Saves calculation time and of course only works backwards.
Retrograde analysis is more efficient than analyzing from start to finish. This is how chess endgame tablebase work. Chess tablebase exist for when only 6 or 7 or 8 pieces are on the board so they start with what they know for sure in the endgame. Then gradually branch out towards the beginning where they end analysis.