I'm converting my comment to an answer. Let $\pi:X\to Y$ be a Galois étale cover, with Galois group $G$. One has a Hochschild-Serre spectral sequence $$E_2 = H^p(G, H^q(X_{et},\mathbb{G}_m))\Rightarrow H^{p+q}(Y_{et}, \mathbb{G}_m)$$ (The reference is given in "A User"'s comment.) The associated exact sequence of low degree terms reads $$0\to H^1(G, H^0(X, \mathbb{G}_m))\to Pic(Y)\to Pic(X)^G \to H^2(G,H^0(X, \mathbb{G}_m))\to Br(Y)$$ where I'm using $Br$, a bit inaccurately, for the 2nd étale cohomology with coefficients in $\mathbb{G}_m$. In fact, the image should lie in $Gr^2Br(Y)$ wrt the filtration on the abuttment. And this seen to be $\ker[\ker Br(Y)\to Br(X)]\to H^1(G,Pic(X))$.

In the affine case, this is surely the same as the sequence of Chase et. al. you gave, but it would need to be checked.