Linfoot produced a bold ride in the opening race and was in position for his first race win of his career until he was overtaken by reigning champion Shane Byrne.
The 29-year-old still finished the race in second to make a return to the podium in what has been a disappointing season so far.
Having set the fastest lap in finishing second, Linfoot went off from pile in race two but his race was cut short by a crash on lap four.
It meant the Knaresborough racing star was unable to claw back towards the crucial end of season Showdown places and he remains tenth in the standings.
“I’ve got the taste of the podium now, so will be keen to get back there soon,” he said.
Linfoot has struggled for consistency since the introduction of the new Honda Fireblade machine at the start of the 2017 season.
He has made the end of season showdown – which creates a six-way shoot-out for the British title - for the last three seasons at Honda.
However, his participation in the finale is under threat – with a 39-point gap to overcome on the top six.
Teammate Jason O’Halloran has made the podium on three occasions and sits fourth in the standings.
But it was Linfoot who shone at Brands Hatch and rose from third on the grid to ride out a thrilling battle with Byrne in a dramatic race.
Byrne has been a regular nemesis for Linfoot on the track and got his bike nose in front once again to deny the Yorkshireman.
Linfoot, who will next race at Thruxton, reflected: “The weekend didn’t really turn out the way I wanted it to in the end.
“Race one was great, I scored my first podium of the season and felt really good with the Fireblade and my confidence was up.
“Setting the fastest lap and starting from pole was also positive, so we have that under our belts now.”
On the crash in race two, Linfoot added: “There’s not much to say really, of course I am disappointed.
“I didn’t think I did anything massively different to what I had before, but it’s just one of those things.
“In hindsight maybe I shouldn’t have tried to push and just settled with fourth or fifth to get some more points, but hindsight is a wonderful thing.”