A method of constructing a self-inflating tire includes: forming a pre-cure green tire body; positioning an air passageway body within the green tire body to extend between an outward facing green tire body surface and the tire cavity, the air passageway body having an enclosed air passageway extending between a hollow body outlet end facing the tire cavity and a hollow inlet end facing outward from the outward facing green tire body surface; curing the green tire body into a finished cured tire body; affixing one or more tapping device(s) over one or both ends of the air passageway body; and establishing through a hollow protrusion member of the tapping device an air flow path extending between the air passageway within the air passageway body and a vent opening within the tapping device. The method includes a routing of the air passageway body through the green tire body along a non-linear path between an outward tire body surface and the inward tire body surface; wherein the non-linear path to at least partially surround a first tire body bead core; and wherein the air passageway body is positioned between overlapping green tire components such as first and second tire turn-up chafer components.