The Suzuki rider led 16 of the first 17 laps on Sunday before handing over the lead to teammate Joan Mir, who took his first MotoGP victory.
This enabled Mir to take a 37-point championship lead with only 50 remaining from the last two races, with Rins now and Fabio Quartararo second in the overall standings.
As an outsider in the championship, 32 points behind Mir ahead of the European GP on Sunday, Rins has not given up his hopes despite the poor math.
“The math says 50 [points remain], these are the points at stake, ”said Rins.
“Joan is having a great season with lots of podiums, but why not? [go for the title still]?
“We’ll keep chopping until the end.”
Rins lost the lead to Mir on lap 17 when Mir was overrun into the right-hand bend at Turn 11 – a result of which he had shifted into first gear instead of second.
From there, Mir was able to use his lead of over a second at a time and keep Rins in check until the checkered flag.
“Well, I think it’s great to be on the podium again,” said Rins after his fourth podium result in five races.
“We got 20 good points for the championship. Let’s look at the last two races now, but I’m pretty comfortable with the bike.
“We did a really good job in many parts of the race and finished in second place.
“Of course I would have liked first place, but I made a small mistake with the gear at Turn 11 and that cost me first place.
“I tried to keep that gap with Joan, but he was able to drive a little faster compared to me. But we are on the right track. “
Suzuki is now on the verge of claiming its first MotoGP crown since 2000 with Mir, but Rins says there have been no discussions about team orders and both riders have simply been warned to avoid “ugly fights” in recent races.
“In the end, they made it clear to us: you have to be respectful on the track, with no ugly fights or elbows,” he added.
“But there we are, arithmetically we’re still fighting for the World Cup.”
Suzuki’s first one-two in the premier class since 1982 means the Japanese brand will also hold the lead in the team and constructors’ standings and Honda will pull all three titles it won last year.