In this paper, we propose an adaptive routing algorithm for vertically partially connected 3D NoCs to (1) overcome failures in vertical links, and (2) find the nearest available vertical link for rerouting of packets. To track the position of each vertical link and distance to the other nodes, the proposed routing algorithm, named Advertiser Elevator, indexes each vertical link and implements a mechanism for announcing and sharing these indexes with the other nodes of the network. Packets are routed toward the nearest vertical link based on received indexes. The routing algorithm tolerates vertical link failures by interpreting the absence of index messages from a vertical link node as a link failure at the node. Packets are rerouted around failed links based on collected messages. The performance of the Advertiser Elevator routing algorithm is evaluated using the Access Noxim NoC simulator under different network congestion levels and fault rates. The results show that the proposed routing algorithm (1) is able to deliver packets as long as there are at least four live vertical links in the network (e.g., corner links) and (2) improves the average network latency by 15% over the well-known Elevator-First routing algorithm.