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 $X(K)\mathrm{\ne }\mathrm{\varnothing }$ for some extension $K\mathrm{/}k$ with $n:=[K:k]\ge 2$ and $\gcd (n,d)=1$, then $X(L)\mathrm{\ne }\mathrm{\varnothing }$ for some extension $L\mathrm{/}k$ 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 $L\mathrm{/}k$ explicitly as well. As an application, we improve upon a result by Coray on smooth cubic surfaces $X\subset {\mathbb{P}}_k^3$ by showing that if $X(K)\mathrm{\ne }\mathrm{\varnothing }$ for some extension $K\mathrm{/}k$ with $\gcd ([K:k],3)=1$, then $X(L)\mathrm{\ne }\mathrm{\varnothing }$ for some $L\mathrm{/}k$ with $[L:k]\in \{1,10\}$.