Zane had rushed out the door as quickly as possible. The simple reality was he knew exactly what Olivia was up to, trying to provoke a rise out of Elena, who also appeared to have feelings for Zane. But Zane was what one might call emotionally stunted, and did not have the capacity to properly deal with a situation like what had transpired in their current camp.
He would rather go out and take on the dangers of the world on his own, then stay in their temporary sanctuary and deal with the drama that women so often caused when around one another.
Zane had scavenged for several hours, finding another carton of cigarettes, as well as some more liquor. He also managed to take down a small group of undead who had a howler embedded in their ranks.