Is your new radiator the OEM or a replacement brand? It could also be that your water pump is leaking but not overt and leak dries up immediately with the warm engine. You mentioned the first time that your car overheated and it died along the way. this is not a good sign and could probably mean a warped or cracked cylinder head. check your dipstick if your engine oil is "milky" in character. Oh, in most toyota vehicles that i've seen and owned, i never saw a radiator cap with a 1.1 rating...