There is a separation axiom between $T_1$ and $T_2$ that Aull (Separation of bicompact sets, <i>Math. Annalen</i> <b>158</b> (1965), 197&ndash;202) calls $J_1$ and that Mukherji (On weak Hausdorff spaces, <i>Bull. Calcutta Math. Soc.</i> <b>58</b> (1966), 153&ndash;157) calls $T_2'$; namely, "every compact subspace is closed."  Such spaces are sometimes called "weak Hausdorff spaces," although this term is sometimes used to mean something else.  See Hoffmann (On weak Hausdorff spaces, <i>Arch. Math. (Basel)</i> <b>32</b> (1979), 487&ndash;504) for further discussion.