This paper generalizes the concept of locally connected graphs. A graph \(G\) is triangularly connected if for every pair of edges \(e_1, e_2 \in E(G)\), \(G\) has a sequence of \(3\)-cycles \(C_1, C_2, \ldots, C_l\) such that \(e_1 \in C_1, e_2 \in C_l\) and \(E(C_i) \cap E(C_{i+1}) \neq \emptyset\) for \(1 \leq i \leq l-1\). In this paper, we show that every triangularly connected \(K_{1,4}\)-free almost claw-free graph on at least three vertices is fully cycle extendable.