There is a theorem of Rogers that for large d, $d$, if you want to cover a ball of radius R $R$ with balls of radius $r \lt R$ then the volume ratio estimates is almost sharp. (Almost = a polynomial expression in $d$; while the volume ration is $(R/r)^d$.)
Morally, the same should be true for caps whether you want to cover large caps by smaller caps or the whole sphere by smaller caps. I think this is also a consequence of the result about covering density that Henry mentioned. There is some basic difference between covering and packing in that coverings are much more efficient than packings. But I cannot say I understand the conceptual reason.
When you want to cover by caps which are very close to being half-spheres then again to the best of my memory the problem becomes delicate. You need always d+1 $d+1$ by Borsuk-Ulam theorem and the precise smallest radius for which d+1 $d+1$ suffices is also not known to the best of my memory.