We can assume that the set $\{p_1,\dots,p_n\}$ is maximal. Note that the set $$K_{n,n-1}=\{\,x\in M\mid |x-p_i|\ge\tfrac\pi2\ \text{for}\ i<n-1\,\}$$ is convex and it contains a point $z$ such that $|p_i-z|=\tfrac\pi2$ for any $i<n$. Since $\{p_1,\dots,p_n\}$ is maximal, $|p_n-z|\le \tfrac\pi2$
Flow $z$ in the gradient flow for $\mathrm{dist}_{p_{n-1}}$ for short time. We get a point $z'$ which will satisfy all your conditions for $\eta_{n,n-1}$.