You have drilled a hole for a door handle and now discover you need a larger hole. One method is to completely plug up the old hole so that the pilot bit of the hole saw has something to centre into. One of our viewers sent in the idea of taking a dowel the size of the existing hole and simply putting it on the end of the pilot drill. This will stabilise the hole saw enough to get a good cut with the larger saw.
This problem is apparently common enough that Starrettt makes a special hole saw arbour that has no pilot hole but rather allows you to mount two holes saws on the same arbour. The first one will be the size of the existing hole, and the second, the size of the hole you want. So now you get a larger hole perfectly centred around the original hole. They call this arbour "Oops". That makes sense.