My sister once lost her wedding ring ring while snowshoeing in deep, fresh, powdery snow at Willamette Pass in oregon (There is a post somewhere on here about that). After searching with SAR for it for hours, declared it lost--and posted signs everywhere and gave SAR the description.
Would you believe that months later, when the snow melted out, someone found it, and actually returned it while not accepting any reward?!
So post some notes at trailheads, tell the rangers around there, etc.--it's possible you will be fortunate as well.