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.