Payment Channel Networks (PCNs) have flourished as one of the most promising solutions to the blockchain scalability problem. Unfortunately, the existing PCN solutions either fail to provide path privacy guarantees or require the not-always-true All-Anonymous-Connected assumption (i.e., an anonymous communication channel always exists for any two participants). To alleviate these problems, we first present a new cryptographic primitive named anonymous multi-hop payment (AMHP), which is an improvement of anonymous multi-hop lock (AMHL). Using AMHP and payment channels, we can have a new PCN solution with path privacy but removing the All-Anonymous-Connected assumption. After that, we present the first AMHP scheme, called AMHL+, by adapting the generic construction of AMHL, but at the cost of high communication overhead. To reduce the communication cost, we further present a new AMHP scheme (named EAMHL+) using bilinear pairing. The communication cost of the EAMHL+ is reduced by 92.3% compared to the AMHL+. The rigorous security analysis demonstrates that the EAMHL+ holds consistency, balance security, and path privacy. Finally, we implement the proposed AMHP schemes using Java. The extensive experimental results show that, though the EAMHL+ requires more computational cost than the AMHL+, it is more efficient than the latter in terms of communication overhead.