Here are my thoughts based on my own experiences with this type of issue:
1) When plunging, use a wet rag or something similar and stuff it in the sink overflow hole and also hold it over the sink overflow hole to try to close it off while plunging. It is hard to do that with one hand while also plunging the drain with your other hand. But, I do my best. If you have a helper, let the helper hold the wet rag in and over the sink overflow hole so you can plunge with both hands. 2) Bend the spring tip of the auger at a slight angle. Then, while you are turning the auger and pushing to advance it, it has a better chance of being able to make any 90 degree turns, including the Tee that you are dealing with. 3) I have a chrome P-trap (not S-trap) under my bathroom sink. In my case, I used a hacksaw (or I could have used a tubing cutter or angle grinder) to cut the horizontal part of my P-trap in two places about an inch apart. Then I put in a chrome coupling that I could easily remove by hand when needed. I used something like this as the coupling:In my case, I have a bathroom sink that leads to a narrow lead pipe drain line under the floor and that has a couple of sharp 90 degree turns, so it is hard to snake it out.