The results of the present article extend the results of [Zbl 1173.11025]. The main result of the article is Theorem 1.1 below. The proof is based on a moving lemma from [Zbl 1188.14015], a recent improvement due to O. Gabber of de Jong's alteration theorem, and the main theorem of [Zbl 1206.13008]. A purity theorem for quadratic spaces is proved as well in the same generality as Theorem 1.1, provided that is local. It generalizes the main purity result from [Zbl 0980.11025] and it is used to prove the main result in [Zbl 1126.14053].