Pirates impressed in Africa by reaching the CAF Confederation Cup final in November, but find themselves in 13th place on the league table following a 1-1 draw with Kaizer Chiefs on Saturday.
The Sea Robbers have 17 points from 15 games - 15 points less than Bidvest Wits in third spot.
"We need to finish much higher in the league and we can do that. There is no big gap between ourselves and the top four,” Tinkler says.
“If we can start grinding positive results behind us, maybe two or three wins, we will climb the table very quickly. That should be an objective right now.
"We have three games coming: Ajax Cape Town, University of Pretoria and then Mamelodi Sundowns. All three of them are very important to collect as many points as possible to ensure that we climb the table.”
Tinkler wants Pirates to improve on their fourth-place finish of last term.
"We shouldn't be finding ourselves in the situation we currently are, out of the top eight. The minimum should be to finish in the top eight.
"Personally I will like to finish better than last season which was fourth. We still have that possibility every season we need to look to bring better performance than what we did in the previous season.
"That's the target we need to set ourselves now. To ensure that we finish with more points than we did last season."