we can place $\sigma$-compact and Lindelöf in the first chain of relations?
No, because it would collapse the chain, since each Tychonoff Lindelöf pseudocompact space $X$ is compact. Indeed, by [Eng, 3.8.11] the space $X$ is paracompact, that is each open cover of the space $X$ has a locally finite open refinement. By [Eng, 3.10.22] the space $X$ is feebly compact, that is if each locally finite family of non-empty open subsets of the space $X$ is finite. We can conclude that the space $X$ is compact.
PS. A stratification of different compact-like spaces is much more subtle, see, for instance, basic [Eng, Chap. 3] and general works [DRRT], [Mat], [Vau], [Ste], [Lip]. The relations between different classes of these spaces are well-studied, some of them are presented at the following diagrams: [M, D:3, p.17], [D-AS, D.1, p. 58](for Tychonoff spaces), [Ste, D. 3.6, p. 611], and [GR, p.2].
[BR] T. Banakh, A. Ravsky. Verbal covering properties of topological spaces, Topology Appl. ** 201** (2016), 181–205.
[D-AS]A. Dorantes-Aldama, D. Shakhmatov, Selective sequential pseudocompactness, Topology Appl., 222 (2017), 53–69.
[DRRT] E.K. van Douwen, G.M. Reed, A.W. Roscoe, I.J. Tree, Star covering properties, Topology Appl., 39:1 (1991), 71-103.
[Eng] R. Engelking, General Topology, Heldermann Verlag, Berlin, 1989.
[GR] O. Gutik, A. Ravsky, On old and new classes of feebly compact spaces, Visnyk of the Lviv Univ. Series Mech. Math. 85 (2018), 48–59.
[Lip] P. Lipparini, *A very general covering property”.
[Mat] M. Matveev, A Survey on Star Covering Properties.
[Ste] R. M. Stephenson Jr., Initially $\kappa$-compact and related compact spaces, in K. Kunen, J. E. Vaughan (eds.), Handbook of Set-Theoretic Topology, Elsevier, 1984, P. 603-632.
[Vau] J. E. Vaughan, Countably compact and sequentially compact spaces, in K. Kunen, J. E. Vaughan (eds.), Handbook of Set-Theoretic Topology, Elsevier, 1984, P. 569-602.