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This is a bit unusual.
Try this -
Find a screw and a washer to match it. Drill a small hole the size of the screw's shank into the stopper (rubber). Make sure that the hole will not be too large and that it is in the center of the stopper. The screw's threads need to be able to grip into the stopper. It needs to be on the side that used to be outside of the stopper.
Attach a piece of fishing line to the washer an put the washer on the screw. Drive the screw into the stopper. Put oil on the stopper. You may be able to pull the string and the stopper out of the salt shaker now.
If this does not work, then put a bit of water inside the salt shaker and slowly heat the shaker up while you pull on the string and stopper. You will need to get the shaker fairly hot so you will need to hold it with some type of insulated mitten while you pull the string. Be sure you heat the shaker very gradually.
I'll check back in 6 hours or so. Please let me know if you would like more information or clarification.
You're welcome and thank you for the "accept". Please let me know how you make out. I am curious if you will need to heat the shaker. You can post back to this question.