Yes and no. 63's only used 19" radiators but 64 and later cars had 22" openings. So it takes a little work to fit a later radiator in a 63. You'll notice that the mounting flange on one side of the later radiator is probably 3" wider than the other. If you don't mind cutting it down, and drilling new holes, it can probably be made to fit. You could possibly drill new holes in your radiator support (passenger side) as well.
If you are having your 63 radiator re-cored, I think a later core can be adapted to your original "bowl" tanks.