An assembly is provided for rotational equipment. This assembly includes a seal device, a ring structure and a fastener. The seal device includes a plurality of seal shoes, a seal base and a plurality of spring elements. The seal shoes are arranged around a 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 base. The ring structure is axially adjacent the seal device. The ring structure includes a fastener aperture extending axially through the ring structure. The fastener is mated with the fastener aperture and abutted axially against the seal device.