I want to re-clarify.
Jax is my main. I'm not looking for help or tips on how to play as Jax against Jax counters. I'm not a perfect Jax player, but I feel extraordinarily comfortable playing as him, and don't have extraordinary problems against any champion.
What I'm looking for are champions that are good against Jax, champions that most Jax players struggle against. And I'm looking for a list of ideas for champions only for the rare scenarios where I'm in a draft mode game and the enemy team has already selected Jax.
This is the only situation in which I'd play anyone but Jax top lane. And in this situation, I'd almost certainly be going against a top lane Jax (there are occasions where they may be playing jungle Jax).
So if the enemy is going to take a top lane Jax and prevent me from playing my strongest champion, then my best bet, if I'm going to go top lane, is to get as good with a Jax counter as I am with Jax.
Then, I'll either be playing with my strongest champion, Jax, against whoever, or I'll be playing against a champion that I'm really good with against Jax, and that other champion is a champion that does particularly well against Jax.
I've played Darius, Jayce, Renekton, and Riven, and I like all 4 (though Riven is the only one I currently have unlocked).
I have Kayle unlocked but have never really tried her.
I've never tried Olaf and I don't have him unlocked.
I have Shen and Malphite, don't particularly love either (don't like them as much as the 4 I mentioned above).