Normally you don't really want to rely on a press fit for axial location of a shaft, normally you would use snaprings, threaded fasteners, flanges and/or shoulders or shaft collars ( https://www.mcmaster.com/6435k15 ) . But under some conditions it might work OK with a press fit.
There are no stock bearings that are undersize, if it says 0.625 in the book, it's going to be 0.625 +/- 0.0001 or better.
In this case I would use High Locking Strength LocTite (Green) https://www.mcmaster.com/loctite to anchor things together, but it depends on how much axial load is on the shaft.