Quantum emitters interacting with photonic band-gap materials lead to the appearance of qubit-photon bound states that mediate decoherence-free, tunable emitter-emitter interactions. Recently, it has been shown that when these band gaps have a topological origin, like in the photonic Su-Schrieffer-Heeger (SSH) model, these qubit-photon bound states feature chiral shapes and certain robustness to disorder. In this paper, we consider a more general situation where the emitters interact with an extended SSH photonic model with longer-range hoppings that displays a richer phase diagram than its nearest-neighbor counterpart, e.g., phases with larger winding numbers. In particular, we first study the features of the qubit-photon bound states when the emitters couple to the bulk modes in the different phases, discern their connection with the topological invariant, and show how to further tune their shape through the use of giant atoms, i.e., nonlocal couplings. Then, we consider the coupling of emitters to the edge modes appearing in the different topological phases. Here, we show that giant-atom dynamics can distinguish between all different topological phases, in contrast to the case with local couplings. Finally, we provide a possible experimental implementation of the model based on periodic modulations of circuit QED systems. Our paper enriches the understanding of the interplay between topological photonics and quantum optics.