I recently had a situation where I needed to enlarge a previously drilled hole. Since I couldn’t use my hole saw’s pilot bit to center the hole, and I don’t have a drill press, I had to come up with a solution. What worked for me was to connect two hole saws together. One was the size of the existing hole. This saw guided the other hole saw that was the size of the hole I needed to drill.
The drawing shows how I did it. The result was a perfectly centered hole with no fuss. The 1⁄4" pilot bit for my hole saws is held in place with a set screw. So I removed the bits from both saws and secured a short length of 1⁄4" steel rod between them.