Last summer, during the shochugeiko, however, we all spent much of the time experimenting with additional counters, or rather, counter-to-the-counter-to-the-counter, and so on, which was a really enlightening experience. Fortunately, much of what was developed was recorded and is also available on the forum.
Today, we worked on aigamae ate and mostly what's presented here:
Basically, there's 1) aigamae ate, countered by 2) picking off the hand to oshi taoshi, which is then countered by a sharp turn into 3) tenkan oshi taoshi. After that, we found a few interesting possibilities you might play with.
First off, we had one pair pick the hand off the wrong way. Well, not the "wrong" way, exactly, but maybe a less efficient way. Instead of tori getting his free hand behind uke's wrist and pushing it, he grabbed in front and pulled. This allowed his other arm to turn palm up toward uke's chin in a form of gyakugamae ate.
We had another pair who, instead of doing the second oshi taoshi, began with the same motion and ended up 1) stepping underneath the arm for tenkai kote hineri, or doing a hip switch and catching 2) sumi otoshi.
Following that, we played with a counter to the second oshi taoshi, or tenkan oshi taoshi. As it's coming on, if tori steps reeeeeeally far out with his outside leg, and pre-turns his foot, he can extend tori out. At that point, tori can reach out with his free hand and snag an "short cut" tenkai kote hineri (where you don't spin under uke's arm).
It's been fun and enlightening to play with and I can't wait to explore more.