The forward is Alan Archibald's only signing of the summer so far after completing his switch from newly- relegated Hearts last month.
Thistle impressed on their return to Scotland's top flight last term and eventually finished tenth, but it could have been so much better had a number of points not slipped through their grasp.
Confidence, however, is high at Firhill that the Jags can improve next time out and Stevenson has lofty ambitions as he gears up for the big kick-off.
He said: "Last season Partick did really well, but you can have a second-season syndrome, which you want to stay away from.
"It is very open and with the players we have got here there is no reason why we can't push up the table.
"There is no reason why we can't be in the top six and push even higher than that. I wouldn't have come here if I didn't think that.
"I never came here for a relegation battle. I know we have got good enough players to be in the top six."
With the new Premiership campaign looming, Thistle have returned to action at their Garscube base after spending a week in the Spanish sunshine.
Stevenson was speaking at the launch of the Jags' new away kit and hours before Motherwell, Aberdeen and St Johnstone kicked off their Europa League second qualifying round clashes last night.
And the new Firhill star reckons Thistle could one day follow in their top-flight counterpart's footsteps.
He said: "I have sampled it and it is a unique thing to be involved in when you are going away to these places and the buzz about it. As much as we didn't have a great run in Europe at Hearts, it was still a great thing to be part of.
"At Hearts, we were expected to be there. But for St Johnstone and Motherwell to be doing it is a massive thing. So we look at those clubs and that is where you want to get to.
"You want to be an established SPFL team and push into those places. When I made my decision, it was because Partick were going in the right direction.
"As a club, we are probably bigger than some of the clubs that are doing it at the minute. There is no reason why we can't do it."
Stevenson and his new team-mates were joined in Spain by defender Darren Barr as the former Hearts and Kilmarnock player attempts to win a deal with Thistle.
And the 29-year-old is keen to be reunited with the stopper if a deal can be done to bring him to Firhill this summer.
He said: "I am good friends with Darren and I have played with him for years. He is a top player and he would be a great addition to the squad, but it is down to the boss and Darren.
"He would be a good signing for us. He is a good lad so hopefully something will get sorted out."