You can do the rehitching ceremony ANYTIME at the Garden Grill, including the Ice Cream Social.
Call Disney Dining to make a PS and be sure to ask them to note on your PS that you want to do the rehitching ceremony. (Maybe just ask them to write "Rehitching Ceremony.) The CM will probably not know what you are talking about, but that's OK. Ask them nicely to just add the note for you. Highly recommend a PS for this -- it just helps make another note about the ceremony.
When you check-in at the Garden Grill, tell the CM on duty that you want to do the rehitching ceremony. You might need to emphasize this because some people asked and apparently it wasn't noted on their card or their server forgot. Anyway, be sure to mention it again, even if it's on your PS.
Tip: Ask if Sam (CM) is working when you are there. Sam is a terrific server; he did our ceremony and he is a great "ham." He managed to snag 3 or 4 extra servers, plus Chip and Pluto to be in our "rehitching party!"
The ceremony is nice. Your server has you stand and he asks each person if you will take the other and you say (or you're SUPPOSED TO SAY) "I do." Then you kiss to the count of 100! It's not that long -- they count oto 10 by 1's, then 10 to 100 by 10's. Hopefully, your server will get everyone within earshot to join in the counting.
After that, you get a nice cake (serves several people easily) and a nice certificate. The cake is complimentary (or at least it used to be).
One more tip: bring your camera or video camera to record the event. It is a terrific magical moment and it is even better if your spouse doesn't know what's coming!

It can be a terrific Disney surprise!!!