That's possible but not certain. If you had a pile of ice cubes on a grate, such that water could drain away, and warm air could filter through accessing that full surface area, that might result in higher heat transfer than the same ice in a bucket full of melt water where the surface area accessible to the air is the surface of the bucket and the surface of the water, and is smaller than the total surface area of all of the cubes.

But if you froze the ice in the bucket to get a roughly cylindrical block of ice, and put that on the grate with the water draining away, the surface area would start out similar to that of the bucket, but would gradually decrease as the ice melted, whereas if you kept it in the bucket with the melt water, the surface area would stay constant.

Topics left as an exercise for the reader are consideration of heat transfer by radiation and the possibility of putting it in a bucket with a hole in it such that the water drains out. And then the challenge of how to fix a bucket with a hole in it without access to a working bucket, which might be important for getting water necessary to sharpen tools for carrying out the repair operation.