For a real number $α\in[0,1]$ and a $k$-uniform hypergraph $\mathcal{H}$, $\mathcal{A}_α(\mathcal{H})=α\mathcal{D}(\mathcal{H})+(1-α)\mathcal{A}(\mathcal{H})$ is called the $\mathcal{A}_α$-tensor of $\mathcal{H}$, where $\mathcal{D}(\mathcal{H})$ and $\mathcal{A}(\mathcal{H})$ are the degree tensor and adjacency tensor of $\mathcal{H}$, respectively. The sum of the $d$-th powers of all eigenvalues of $\mathcal{A}_α(\mathcal{H})$ is called the $d$-th order $\mathcal{A}_α$-spectral moment of $\mathcal{H}$, which is equal to the $d$-th order trace of $\mathcal{A}_α(\mathcal{H})$. In this paper, some hypergraphs are ordered lexicographically by their $\mathcal{A}_α$-spectral moments in non-decreasing order. The first, the second, the last and the second last hypergraphs among all $k$-uniform linear unicyclic hypergraphs and hypertrees are characterized, respectively. We give the first and the last hypergraphs among all $k$-uniform linear unicyclic hypergraphs with given grith, and characterize the last hypertree among all $k$-uniform hypertrees with given diameter. Furthermore, we determine some extreme values of the $\mathcal{A}_α$-spectral moments for hypertrees and linear unicyclic hypergraphs, respectively.