A hole saw is completely ineffective when used to enlarge an existing hole. Here's why: When you're drilling a new hole, the pilot bit guides the saw into the wood and keeps it cutting on course. But when you're enlarging a hole, there is no wood for the pilot bit to bore into. As a result, the hole saw wobbles recklessly all over the place.
To overcome this problem, fasten a piece of 1/2-inch plywood over the existing hole. Then bore into the plywood and through the hole behind it. The plywood will hold the pilot bit on track until the saw starts cutting.
When enlarging a lockset hole, mark the centerlines of the old hole on the door. Then mark centerlines on the plywood block. Lay the block on the door and align the two sets of center marks. Place a second plywood block behind the door and clamp the parts together, as illustrated. Set the pilot bit on the intersecting centerlines and bore clean through the plywood and door. The second plywood block will prevent the hole saw from splintering the back of the door.