learning rate very low 1e-5 for Adam optimizer good practice?
I think that for the most part, the ends justify the means when it comes to learning rates. If the network is training well and you're confident that you're evaluating its generalization properly, use what works. 

With that said, overfitting isn't usually caused by high learning rate.  It's hard to say with the information you've supplied here, but I suspect that your regularization isn't set up well enough, so using a high learning rate overfits the poorly regularized network rapidly, while training with a low learning rate doesn't get close to a local optimum because it can't move fast enough, but it's still heading towards an overfit optimum.

I'd check in on your regularization. What are you using?