Can’t light the Menorah till Temple is cleansed?

I would like to know how they were able to immediately light the menorah before the Temple was cleansed? I thought it takes eight days to re-dedicate the Temple, and only then could you light the menorah (and place the showbread and burn the incense on the golden altar). Can someone clear this up by giving me the laws and the timeline of the Temple rededication at that time? Also, I assume there was olive oil in Israel beside that one ‘kosher’ cruse/jug of oil. Why couldn’t they use that on the second day? If you say the oil needs [eight days] to be sanctified, it seems like it’s OK to light a menorah with sanctified oil immediately, but the rest of the Temple needs eight days to sanctify. If this is so, then the menorah must have been lit outside the polluted Temple for eight days and then brought back in after the dedication process.