When do connected spaces have nice connected preimages?

We prove that every connected Tychonoff space is an open monotone continuous image of a connected strictly $\sigma$-discrete left-separated Tychonoff space. For wide classes of connected spaces it is established that they have a finer Hausdorff strictly $\sigma$-discrete connected topology. Another result is that a finer Tychonoff connected strictly $\sigma$-discrete topology exists for any Tychonoff topology with a countable network. We show that there are Tychonoff connected spaces with countable network which are not continuous images of connected second countable spaces. It is established also that every connected Tychonoff space $\mathcal{X}$ is an open retract of a connected homogeneous Tychonoff space, while it is not always possible to find a finer connected homogeneous topology on $\mathcal{X}$.