Using the old race works well, but I would recommend that you don't use it "backwards." The old race will typically seat to some degree; if it's in backwards it's tough to knock out, as the shoulder is facing the wrong way.

To remove the inner race, I use a large 1/2" drive socket installed backwards on an extension bar. The larger circumference of the socket shoulder (as compared to drift) gives a better grip on the shoulder of the bearing cup.