There are at least 3 possibilities for creating the onion ring join using the tatting needle.
1. I consider this to be the easiest method. The inner ring with its picot is already completed at this point. It will be flopping loose below the needle. Slide the work on the needle close the needle's point to give you greater access to the inner ring. Then just slide the picot of the inner ring over the point of the needle and complete the ring as normal. The picot on the inner ring will twist a bit.
2. This method is the closest to the regular shuttle tatted onion ring join procedure. Follow the directions for the shuttle tatted onion ring join but with these changes. Wrap the double stitches (ds) needed for the first part of the ring onto the needle up to the join spot; slide the ds off the needle but do not close the ring. Next make the join using the point of the needle to pull the ball thread up through the picot and slide it onto the needle. Continue wrapping the needed ds until ring done. Now slide all the ds off the needle and gradually bring all double stitches together to close the ring.
3. The most sensible method of creating the onion ring join effect is to make the inner as a normal needle tatted ring and then surround the inner ring with a chain joined to the picot of the inner ring with a shuttle lock join. The procedure is to wrap the ds for the first inner ring and close it. Leave no bare thread space and wrap the ds needed for a chain and slide it off. Now make a shuttle lock join or make a join using the ball thread. Finish the remainder of the chain. When done, run the point of the needle through the bottom of the inner ring to anchor it or make a lock join to the bottom of the ring.
Using the shuttle lock join with the chain method will make a "dimple" at the join. The ball thread join will make a join more like the true onion ring join.
Substitute a tatting needle for the shuttle in the drawing of the shuttle lock join.
In this example of a shuttle tatted onion ring join, imagine the core thread which goes through the loop brought up through the picot as being the tatting needle.