I found/figured out the solution.
Starting with the reference to this problem here:
A few entries down the page, a more illuminating further reference is given:
This latter entry shows the crucial mating position: White knights on g3 and e0, ...
Here is an upper bound. Since is has been proven that 8 pieces, with opposite colored bishops since the question asks for starting pieces only, that only 63 squares can be covered. Addding a single pawn works since occupied sqaures count as covered.
I say upper bound because all of the sqaures occupied by the pieces are also attacked, which is not needed ...
I start by excluding mating positions.
If white King steps to e0 from e1 (from a normal stalemate position) the black king is outside of the red line and once the white king is on e0 there will be no attacking piece to deliver mate. If the position is not the stalemate illustrated above, but with one of the knights attacking, then white king doesn`t need to ...
Maybe the last man is:
(I think this line is forced step by step, that`s why I did not include any further explanation. Obviously after sacrificing all, the last piece left for white is the pawn that eventually mates.)