Stevens made his home debut in their 19-13 loss to Ealing, having joined the club after leaving financially-stricken London Welsh last month.
"We're pushing for top four, we want top four," the ex-Nottingham, Worcester and Plymouth Albion centre said.
"We want to do really well in the British and Irish Cup and we want to do as well as we can."
He told BBC Radio Jersey: "This club needs to be pushed forward and I think they definitely are on the right road for doing that."
The island side are currently eighth in the Championship, but are just seven points behind fourth-placed Ealing and point further back from Doncaster in third.
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New Jersey signing Heath Stevens says the club are capable of reaching the Championship play-offs.