A mouse can climb out of a bucket if you don't make the bucket slippery on the inside.
You can put some olive oil or something on the buckets inside walls halfway up and then the mice will be unable to climb out of the bucket since the sides will be too slippery for the mice.
Fill the bucket halfway with some water when you create the mouse trap so that they will eventually drown and make it harder for the mice to escape the bucket mouse trap.
Make the bucket mouse trap with a spinning bottle that is slippery so that when the mouse climbs onto the spinning bottle the mouse will right away fall into the bucket and be trapped.