Much research has identified and confirmed the core elements of the well-known Community of Inquiry Framework (CoIF): Social, Cognitive and Teaching Presence (Garrison, 2011). The overlap of these Presences, their definitions and roles, and their subsequent impact on the educational experience, has received less attention. This article is prompted by the acceptance of that omission (Garrison, Anderson, & Archer, 2010). It proposes enrichment to the Framework, by entitling the overlapping spaces uniting pairs of Presences as "Influences." These three spaces, linking pairings of Social, Teaching, and Cognitive Presences, can be labelled as "trusting," "meaning-making," and "deepening understanding." Their contribution to the educational experience is to address constructively some of the challenges of online learning, including learner isolation, limited learner experience of collaborative group work and underdeveloped higher-level abilities. For these purposes we also envisage "cognitive maps" as supporting learners to assess progress to date and identify pathways forward (Garrison & Akyol, 2013). Such maps, developed by a course team, describe the territory that learners may wish to explore, signpost possible activities, and encourage the development of cognitive and interpersonal abilities required for online learning. We hope that considering the Influences may also assist tutor conceptualisations of online community-based learning. Our proposals call on both learners and tutors to conceive of the Presences and Influences as working together, in unison, to enhance the educational experience whilst fostering deep learning. Our suggestions are presented to stimulate scholarly debate about the potential of these interwoven sections, constructively extending the Framework.