What's an example of a non-Hausdorff $T_1$-space which is connected, but not hyperconnected?

Question

The standard example of a non-Hausdorff space which is $T_1$ is either a cofinite space over an infinite base set or a cocountable space over something uncountable. However, those topologies are already hyperconnected (we cannot cover $X$ by two distinct proper closed subsets), which implies that not only do there exist two points which cannot be separated by disjoint open subsets, but in fact no two points can be separated in that way (hint: reformulate the separability statement in terms of closed sets).

The obvious „fix“ to give a space that is not hyperconnected would be to take the disjoint union with any other second space, which of course retains the non-Hausdorffness.

But are there any connected examples?

Answer 1

What about the line with two origins? Every point, including the two origins, is closed, but the two origins can't be separated by disjoint open sets. It is the union of two proper nonempty closed sets, but not two disjoint nonempty closed sets, so it is connected but not hyperconnected.

Answer 2

There is another "obvious fix" that does preserve connectedness--take a wedge sum with a second space. If $X$ is any connected non-Hausdorff $T_1$ pointed space and $Y$ is any connected $T_1$ pointed space with more than one point, then $X\vee Y$ is connected, $T_1$, and non-Hausdorff, but not hyperconnected since $X$ and $Y$ are both closed in it.