Brewster's Scottish rink lie second in the standings after a narrow 7-6 win over China that followed a disappointing 5-3 defeat to France.
They share the same 5-2 record as China and Sweden with Canada out front after winning all seven of their matches in Basel.
And Brewster's Scotland, silver medallists last year, will face a crunch day on Wednesday, taking on both Sweden and Canada - their main medal rivals.
"They're a good team. They missed nothing in the first two ends and I really thought we were on a sticky wicket," said Brewster, after the morale-boosting win over China.
"I was disappointed with the result against France and I couldn't get draw-weight - but we're right back in the mix now."
However, hosts Switzerland saw their nightmare championship continue as they prop up the 12-team standings with just one win from seven.