A number of user authentication schemes have been proposed to provide roaming services in the global mobility network (GLOMONET). However, most of these schemes are based on an asymmetric cryptosystem, which has a higher computational cost. To achieve computational efficiency, Chang et al. proposed an authentication scheme using simple hash functions for mobile devices in the GLOMONET. However, this study shows that the scheme by Chang et al. does not provide user anonymity and cannot counteract insider attacks, in addition to being vulnerable to the disclosure of session keys and foreign agent spoofing. Therefore, this study proposes a more secure and efficient authentication scheme and uses the Burrows-Abadi-Needham (BAN) logic method to verify the scheme. The proposed scheme can overcome the main disadvantages of the Chang et al. scheme and satisfy the crucial design criteria for a secure remote user authentication scheme. The proposed scheme can provide a more secure functionality and has superior performance, such as a lower computational cost, less time complexity, fewer communication rounds, fewer transmitted messages, and less energy consumption. For practical purposes, this study demonstrates that the proposed scheme can be used to enhance the effectiveness and efficiency of the authentication scheme in the GLOMONET.