Can you respond to your own spell being countered by casting another spell?

When an opponent casts a counterspell targeting your spell, both spells are on the stack at the same time. Priority is passed after the counterspell is placed on the stack, meaning you receive priority and can cast another spell or activate an ability in response before the counterspell resolves (CR 117.3b, CR 116.3).

However, once the counterspell actually resolves, your original spell is countered and moved to the graveyard (CR 701.5a). At that point, you cannot retroactively respond to the countering itself — the window to act was before resolution, not after.

It's also worth noting that casting another spell in response to a counterspell does not stop the counterspell from resolving. The stack resolves one item at a time, last-in first-out (CR 405.6i). Your new spell will resolve first, then the counterspell will resolve and counter your original spell unless you also cast something to counter the counterspell itself.

Example: You cast Lightning Bolt. Your opponent casts Cancel targeting it. Before Cancel resolves, you cast Negate targeting Cancel. Now the stack is: Lightning Bolt → Cancel → Negate. Negate resolves first, countering Cancel. Then Lightning Bolt resolves and deals 3 damage normally.