FOMOCO experienced severe overheating problems with the 58/59 BIRD and it was finally addressed on 1960 model run with a heavier radiator, five blade fan and shroud (AC equipped).
-1960 BIRD COOLING SYSTEM UPGRADE(S)-
As I am overly anal, I would choose to have the original pump rebuilt (closely inspect impeller condition) if it has the correct Casting I.D. Nos.
A radiator over the years will build scale deposits in the core and the cooling efficiency will suffer, especially if the cooling system has not been serviced properly. If the core appears good (no obvious corrosion/leakage), most any certified radiator shop can perform a flow test. Have it re-cored (MODINE) to keep the OEM appearance.
If severe overheating is a problem (after fan shroud and blade upgrade), consider a trans cooler and a free flowing thermostat.