The current document is directed to methods and systems that implement an application-level tunnel through which syslog messages are transmitted. In one implementation, the message portion of syslog messages transmitted through the application-level tunnel includes both a unique message identifier and a long-message-indication field. Syslog messages that contain message portions of greater lengths than a specified maximum length, referred to as “long syslog messages,” are transmitted through the application-level tunnel as multiple component syslog messages, each with a message portion that includes, in addition to a unique message identifier and a long-message-indication field with a positive value, a long-message field that contains a sequence number for the long-message portion and the total number of long-message portions.