They do sometimes do request hours, especially John Scott. He mentioned that he had one tomorrow - at 5.00 p.m. I think he said. You might be lucky with a request at other times, but John will pull out all the stops to get your item for you in a request hour. The prices are no higher than at any other time.
Rocks & Co also put all the previous days' items (those that aren't sold out) on the website, so if you take a look your ring may well be there. Be warned, though, the prices on the web are MUCH higher than on air, but at least you can get the item number for when you put in a request.