The 35-year-old led his side to a 36-run defeat against the Windies, but admitted that he had made his decision prior to the tournament.
However, the batsman will stay in charge of the one-day and Test match sides for the rest of the year.
"I think we need a young leader," Jayawardene said. "It's a great opportunity for somebody to start in the T20 format.
"I haven't stepped down from the other formats - obviously I took over until December. I will assess what I want to do after that.
"It hurts a lot because you want to do something special, not just personally but for the public as well," he added. "We've been playing some really good cricket, but unfortunately we haven't been able to cross that [final] hurdle."
Sri Lanka host New Zealand later this month.