With the growing usage of Bitcoin and other cryptocurrencies, many scalability challenges have emerged. A promising scaling solution, exemplified by the Lightning Network, uses a network of bidirectional payment channels that allows fast transactions between two parties. However, routing payments on these networks efficiently is non-trivial, since payments require finding paths with sufficient funds, and channels can become unidirectional over time blocking further transactions through them. Today's payment channel networks exacerbate these problems by attempting to deliver all payments atomically. We present the Spider network, a new packet-switched architecture for payment channel networks that addresses these challenges. Spider splits payments into transaction units and transmits them over a period of time across different paths. Spider uses congestion control, in-network scheduling, and imbalance-aware routing to optimize delivery of payments. Our results show that Spider improves the number and volume of successful payments on the network by 10-75% and 10-35% respectively compared to practical state-of-the-art approaches.