Payment channel networks (PCNs) are a promising solution to the blockchain scalability problem. In PCNs, a sender can route a multi-hop payment to a receiver via intermediaries. Yet, Lightning, the only prominent payment channel network, has two major issues when it comes to multi-hop payments. First, the sender decides on the path without being able to take local capacity restrictions into account. Second, due to the atomicity of payments, any failure in the path causes a failure of the complete payment. In this work, we propose Forward-Update-Finalize (FUFi): The sender adds redundancy to a locally routed payment by initially committing to sending a higher amount than the actual payment value. Intermediaries decide on how to forward a received payment, potentially splitting it between multiple paths. If they cannot forward the total payment value, they may reduce the amount they forward. If paths for sufficient funds are found, the receiver and sender jointly select the paths and amounts that will actually be paid. Payment commitments are updated accordingly and fulfilled. In order to guarantee atomicity and correctness of the payment value, we use a modified Hashed Time Lock Contract (HTLC) for paying that requires both the sender and the receiver to provide a secret preimage. FUFi furthermore is the first local routing protocol to include fees and specify a fee policy to intermediaries on how to determine their fair share of fees. We prove that the proposed protocol achieves all key security properties of multi-hop payments. Furthermore, our evaluation on both synthetic and real-world Lightning topologies shows FUFi outperforms existing algorithms in terms of fraction of successful payments by about 10%.