Degrees of closed points on hypersurfaces

Let $$k$$ be any field. Let $$X \subset \mathbb {P}_k^N$$ be a degree $$d \ge 2$$ hypersurface. Under some conditions, we prove that if $$X(K) \ne \emptyset$$ for some extension $$K/k$$ with $$n:=[K:k] \ge 2$$ and $$\gcd (n,d)=1$$, then $$X(L) \ne \emptyset$$ for some extension $$L/k$$ with $$\gcd ([L:k], d)=1$$, $$n \nmid [L:k]$$, and $$[L:k] \le nd-n-d$$. Moreover, if a $$K$$-solution is known explicitly, then we can compute $$L/k$$ explicitly as well. As an application, we improve upon a result by Coray on smooth cubic surfaces $$X \subset \mathbb {P}^3_k$$ by showing that if $$X(K) \ne \emptyset$$ for some extension $$K/k$$ with $$\gcd ([K:k], 3)=1$$, then $$X(L) \ne \emptyset$$ for some $$L/k$$ with $$[L:k] \in \lbrace 1, 10\rbrace$$.