We consider a multi-hop wireless network where energy-harvesting nodes rely solely on ambient radio-frequency interference as their power source. The goal is to deliver a packet from source to destination with minimum latency while guaranteeing a target end-to-end outage probability under Rayleigh fading. This gives rise to a joint routing and power allocation problem in which interference plays a dual role: it provides harvestable energy, but also impairs link reliability. We first transform the intricate, multiplicative end-to-end outage constraint into an additive form. Then, we propose two optimal algorithms. The first, HOPE, minimizes the bottleneck harvesting latency, i.e., the longest energy-accumulation time along the path, by reducing the problem to a least-cost path computation on a graph with interference-aware link weights. The second, THOPE, extends the objective to total end-to-end latency, adding the cumulative transmission delay, and likewise admits a closed-form least-cost path solution. Both algorithms are solved in polynomial time via Dijkstra's algorithm and yield explicit expressions for the optimal transmit powers. Extensive simulations reveal that HOPE and THOPE reduce the average total latency by up to an order of magnitude compared with shortest-path and minimum-energy benchmarks, while exhibiting sharply concentrated delay distributions that are largely insensitive to interferer density.

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