This paper studies the problem of fair distribution of indivisible goods under connectivity constraints. Goods are represented as vertices in a connected graph, and the set of goods allocated to each agent is a connected subgraph of this graph. The paper focuses on the maximin share criterion, a widely studied fairness criterion. It is well known that there may not be an allocation that satisfies the min-max share criterion even without connectivity constraints. Therefore, it is natural to find an approximate allocation that guarantees each agent a connected bundle of goods that is at least a certain percentage of the min-max share value. It is known that such approximate allocations exist for some graph classes, such as complete graphs, cyclic graphs, and d-claw-free graphs for fixed d. However, whether such approximate allocations exist for all graph classes is an open question. In this paper, we continue our systematic study of the existence of approximate allocations for limited graph classes and show that such allocations exist for several well-studied graph classes, including block graphs, cactus graphs, complete multipartite graphs, and partitioned graphs.