Can you sacrifice a permanent to pay an additional cost after a spell has already been countered?

Costs — including additional costs like sacrificing a permanent — are paid during the casting of a spell, before any player receives priority and before any spell can be countered. According to CR 601.2, casting a spell involves announcing it, making choices, and then paying all costs in sequence. Only after all costs are fully paid does the spell become cast and land on the stack.

Once the spell is on the stack and players receive priority (CR 116.3), opponents may respond with counterspells or other effects. If the spell is countered at that point, the costs have already been paid — you cannot go back and 'undo' the sacrifice, nor are there any remaining costs left to pay after countering.

CR 701.5 defines 'counter a spell' as moving it from the stack to its owner's graveyard. This happens after costs were paid, not before. There is no game state in which a spell exists on the stack with costs still unpaid and simultaneously subject to being countered.

Concrete example: You cast Act of Treason and, as an additional cost via a different card's ability, sacrifice a creature. Your opponent then casts Counterspell. The sacrificed creature is already gone — it was sacrificed when you cast the spell. The counter removes the spell from the stack, but does not restore the creature you sacrificed.

There is no scenario where you sacrifice a permanent to pay a cost after a spell is countered, because countering always occurs after costs are paid (CR 601.2h, 701.5).