An assembly includes a plurality of seal shoes, a seal base and a plurality of spring elements. The seal shoes are arranged around an axis in an annular array. 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 to the seal base. A first of the spring elements includes a first mount, a second mount and a spring beam. The first mount is connected to the first seal shoe. The second mount is connected to the seal base and disposed a circumferential distance away from the first mount. The spring beam extends longitudinally along a non-straight centerline between and connected to the first mount and the second mount.