A non-contact seal assembly is provided. This assembly includes a plurality of seal shoes, a seal base and a plurality of spring elements. The seal shoes are arranged about a centerline in an annular array. The seal shoes include a first seal shoe extending axially along the centerline between a first shoe end and a second shoe end. An aperture may extend partially axially into the first seal shoe from the first shoe end and laterally within the first seal shoe. The seal base circumscribes the annular array of the seal shoes. Each of the spring elements is radially between and connects a respective one of the seal shoes with the seal base.