Hair colour in mammals is always a result of various forms of melanin
Eumelanin, pheomelanin, etc.
That pigment just doesn't have a whole lot of range of expression, so your options are limited.
Having other pigments show up in humans would be very difficult to arrange, barring genetic engineering.
Birds have carotenoids (which, however, are acquired from their diet) and Porphyrins. But getting those into humans would surely involve genetic engineering.
Note that even birds don't produce blue pigments - the blue of blue jays and the green of parrots are actually due to interference patterns produced by the structure of the feathers.
Note that hair doesn't have that kind of solidity or structure.
So.... Bottom line. Natural hair colour is probably limited to a range not much greater than currently exists.