In fact there are spaces which are minimal Hausdorff'' -- they have no coarser Hausdorff topology -- but are not compact. It turns out that these spaces are H-closed'' (every open cover has a finite subfamily whose closures cover) and semi-regular (the collection of regular open sets form a base). A minimal Hausdorff space is compact exactly when it is Urysohn. Spaces which have coarser minimal Hausdorff topologies are called Kat\v etov. A ``nice'' example of a space which is not Kat\v etov is the space of rational numbers $\mathbb{Q}$.

I'm not sure about compact spaces, but I suspect that a Hausdorff space has a unique coarser minimal Hausdorff topology exactly when it is H-closed. One direction I'm sure of -- the semiregularization of an H-closed space is minimal Hausdorff.

BTW, (one of) THE BOOK(s) on this topic is Extensions_and_Absolutes_of_Hausdorff_Spaces by Porter and Woods, however it discusses Hausdorff spaces almost exclusively.

In fact there are spaces which are "minimal Hausdorff" -- they have no coarser Hausdorff topology -- but are not compact. It turns out that these spaces are "H-closed" (every open cover has a finite subfamily whose closures cover) and semi-regular (the collection of regular open sets form a base). A minimal Hausdorff space is compact exactly when it is Urysohn. Spaces which have coarser minimal Hausdorff topologies are called Katĕtov. A "nice" example of a space which is not Katĕtov is the space of rational numbers $\mathbb{Q}$.

I'm not sure about compact spaces, but I suspect that a Hausdorff space has a unique coarser minimal Hausdorff topology exactly when it is H-closed. One direction I'm sure of -- the semi-regularization of an H-closed space is minimal Hausdorff.

By the way, (one of) THE BOOK(s) on this topic is Extensions and absolutes of Hausdorff spaces by Porter and Woods, however it discusses Hausdorff spaces almost exclusively.