Back then, the only way I saw to make the hole larger was to epoxy a wooden plug into the existing hole and then drill a new one, with the hole saw pilot bit cutting into that wooden plug. That worked fine but was messy and time consuming (yard time = $$).
As in most cases, whenever you stub your toe you need to view this as an opportunity. Someone has. Multiple someones in fact. Enter the hole enlarger:
This tool is really quite simple - the nut that holds the hole saw on the mandrel is replaced with a nut that has its own mandrel, allowing a second (smaller) hole saw to be mounted inside the main one. Because the fancy nut is made in such a way that the inner hole saw protrudes beyond the primary hole saw, it acts as the pilot. To enlarge a hole, you install the new larger size saw on the primary mandrel, and a saw matching the size of the existing hole as the inner saw. Simple, really, now that you see it.
These are available from Bosch, Vermont American, and even Sears. You should be able to find one for less than $20... I paid $4.95 for mine.