A search was made for possible half-metallic (HM) antiferromagnet (AFM) in all the ($C_2^{29}$=406) double perovskites structures of Sr₂$BB′$O₆ where $BB′$ pairs are any combination of 3d, 4d or 5d transition elements with the exception of La. Sr can also be replaced by Ca or Ba whenever HM-AFM was found and similar calculations were then performed in order to probe further possibilities. It was found that $A$₂MoOsO₆, $A$₂TcReO₆, $A$₂CrRuO₆, where $A$=Ca, Sr, Ba, are all potential candidates for HM-AFM. The AFM of $A$₂$BB′$O₆ comes from both the superexchange mechanism and the generalized double exchange mechanism via the $B$($t$_2$g$)-O2$p$_$π$-$B′$($t$_2$g$) coupling, with the latter also being the origin of their HM. Also considered were the effects of spin-orbit coupling (SOC) and correlation (+$U$) by introducing +SOC and +$U$ corrections. It is found that the SOC effect has much less influence than the correlation effect on the HM property of the compounds. For $A$₂TcReO₆ and $A$₂CrRuO₆, after+$U$, they become nearly Mott-Insulators. In the future, it is hoped that there will be further experimental confirmation for these possible HM-AFM candidates.