Could be on the shock absorber with a leak of pressure inside or something wrong inside that gives a "resonnance" vibration which field is in beetwenn 60 mph and over then stops in higher speed ?
you can check this in doing a test on safe place where you can go at 55 then go at 60 and if shimmy start just brake down hardly and let the car go then ; if the shimmy disapear that 's probably somme issue on front Chock-units
Hope this idea helps ?