An assembly is provided for rotational equipment. This rotational equipment assembly includes a plurality of seal shoes, a seal base, a plurality of spring elements and a secondary seal assembly. The seal shoes are arranged circumferentially about an axial centerline in an annular array. The seal shoes include a first seal shoe. The seal base circumscribes the annular array. The spring elements include a first spring element. The first spring element connects and extends between the first seal shoe and the seal base. The secondary seal assembly is configured to seal a gap between the seal base and the seal shoes. An axial end portion of the first seal shoe projects axially along the axial centerline, in a direction away from the first spring element, beyond the secondary seal assembly.