Former party leader Cassy O'Connor has 37 per cent of the primary vote in Hobart - one of three Legislative Council divisions that went to the polls on Saturday.
It comes after the Greens boosted their lower house representation from two to five MPs at the March 23 election.
The party picked up a 1.5 per cent swing, coinciding with the lower house increasing from 25 to 35 members.
Tasmania is widely acknowledged as being home to world's first 'green' party - the United Tasmania Group which ran candidates at the 1972 state election.
The Greens secured five seats in the lower house at Tasmania's election in March. (Loic Le Guilly/AAP PHOTOS)
Counting in Hobart, as well as Elwick and Prosser, resumed on Monday with final results potentially known in coming days.
Ms O'Connor has a sizeable lead over independent candidate John Kelly (22 per cent) and is expected to prevail.
She resigned from the party leadership in July and left parliament for a tilt at the upper house.
Local mayor and independent candidate Bec Thomas leads Labor's Tessa McLaughlin in Elwick, an electorate covering Hobart's northern suburbs held by Labor.
In Prosser, held by the Liberals, the party's Kerry Vincent is in the box seat ahead of Labor's Bryan Green.
Tasmania's 15-member upper house was made up of seven independents, four Liberal and four Labor members prior to Saturday's vote.
4°C