Summary: A \(\operatorname{mod}\,(2 p + 1)\)-orientation \(D\) is an orientation of \(G\) such that \(d_D^+(v) \equiv d_D^-(v)\pmod{2 p + 1}\) for any vertex \(v \in V(G)\). Extending Tutte’s integer flow conjectures [W. T. Tutte, Can. J. Math. 6, 80–91 (1954; Zbl 0055.17101)], it was conjectured by Jaeger that every \(4 p\)-edge-connected graph has a \(\operatorname{mod}\, (2 p + 1)\)-orientation. However, this conjecture has been disproved in [M. Han et al., J. Comb. Theory, Ser. B 131, 1–11 (2018; Zbl 1387.05107)] recently. Infinite families of \(4 p\)-edge-connected graphs (for \(p \geq 3\)) and \((4 p + 1)\)-edge-connected graphs (for \(p \geq 5\)) with no \(\operatorname{mod}\, (2 p + 1)\)-orientation are constructed in M. Han et al. [loc. cit.]. In this paper, we show that every family of graphs with bounded independence number has only finitely many contraction obstacles for admitting \(\operatorname{mod}\, (2 p + 1)\)-orientations, contrasting to those infinite families. More precisely, we prove that for any integer \(t \geq 2\), there exists a finite family \(\mathcal{F} = \mathcal{F}(p, t)\) of graphs that do not have a \(\operatorname{mod}\, (2 p + 1)\)-orientation, such that every graph \(G\) with independence number at most \(t\) either admits a \(\operatorname{mod}\, (2 p + 1)\)-orientation or is contractible to a member in \(\mathcal{F}\). This indicates that the problem of determining whether every \(k\)-edge-connected graph with independence number at most \(t\) admits a \(\operatorname{mod}\, (2 p + 1)\)-orientation is computationally solvable for fixed \(k\) and \(t\). In particular, the graph family \(\mathcal{F}(p, 2)\) is determined, and our results imply that every 8-edge-connected graph \(G\) with independence number at most two admits a mod 5-orientation.