Little is known about the high-order interactions among brain regions measured by the similarity of higher-order features (other than the raw blood-oxygen-level-dependent signals) which can characterize higher-level brain functional connectivity (FC). Previously, we proposed FC topographical profile-based high-order FC (HOFC) and found that this metric could provide supplementary information to traditional FC for early Alzheimer's disease (AD) detection. However, whether such findings apply to network-level brain functional integration is unknown. In this paper, we propose an extended HOFC method, termed inter-network high-order FC (IN-HOFC), as a useful complement to the traditional inter-network FC methods, for characterizing more complex organizations among the large-scale brain networks. In the IN-HOFC, both network definition and inter-network FC are defined in a high-order manner. To test whether IN-HOFC is more sensitive to cognition decline due to brain diseases than traditional inter-network FC, 77 mild cognitive impairments (MCIs) and 89 controls are compared among the conventional methods and our IN-HOFC. The result shows that IN-HOFCs among three temporal lobe-related high-order networks are dampened in MCIs. The impairment of IN-HOFC is especially found between the anterior and posterior medial temporal lobe and could be a potential MCI biomarker at the network level. The competing network-level low-order FC methods, however, either revealing less or failing to detect any group difference. This work demonstrates the biological meaning and potential diagnostic value of the IN-HOFC in clinical neuroscience studies.