Stockfish already does that. You run it as long as you want. When it's calculating evaluations for move k+1, it already knows what move k was, because it has that evaluation. It can explore that tree first, before exploring other trees to try to improve on move k.

Stockfish never settles down to an exact move, unless it sees a forced mate sequence. You could leave it running indefinitely, but the exponential blowup will soon grind you to a halt.

What new versions of Stockfish buy you are optimizations and better heuristics. An older version of Stockfish can still beat a new one, if given enough of a compute advantage.