Which spaces have a coarser connected Hausdorff topology?

We present some answers to the title. For example, if K is compact, zero-dimensional and D is discrete, then K⊕D has a coarser connected topology iff w(K)⩽2|D|. Similar theorems hold for ordinal spaces and spaces K⊕D where K is compact, not necessarily zero-dimensional. Every infinite cardinal has a coarser connected Hausdorff topology; so do Kunen lines, Ostaszewski spaces, and Ψ-spaces; but spaces X with X⊂βω and |βω⧹X|<2c do not. The statement “every locally countable, locally compact extension of ω with cardinality ω1 has a coarser connected topology” is consistent with and independent of ZFC. If X is a Hausdorff space and w(X)⩽2κ, then X can be embedded in a Hausdorff space of density κ.