This letter presents the first method for autonomous exploration of unknown cavities in three dimensions (3D) that focuses on minimizing the distance traveled and the length of tether unwound. Considering that the tether entanglements are little influenced by the global path, our approach employs a 2-level hierarchical architecture. The global frontier-based planning solves a Traveling Salesman Problem (TSP) to minimize the distance. The local planning attempts to minimize the path cost and the tether length using an adjustable decision function whose parameters play on the trade-off between these two values. The proposed method, TAPE, is evaluated through detailed simulation studies as well as field tests. On average, our method generates a 4.1% increase in distance traveled compared to the TSP solution without our local planner, with which the length of the tether remains below the maximum allowed value in 53% of the simulated cases against 100% with our method.