The 19-year-old, who is the youngest competitor to start the Tour since 1937, finished third on the 155.3km stage from Mulhouse to Le Markstein, just days after his maiden podium at Le Lioran.
The Decathlon CMA CGM rider now sits 15 seconds behind Belgium's Remco Evenepoel, who occupies third in the general classification. Slovenia's Tadej Pogacar and Denmark's Jonas Vingegaard occupy the top two spots.
"It's really a pleasure to have this white jersey and still be in the hunt for the podium," Seixas said after taking the jersey from Juan Ayuso (Lidl-Trek) by a margin of three seconds.
"It's unbelievable, honestly. It's beautifully tough. Today was another challenging day - we're starting to get used to that on the Tour."
Seixas' team showed their strength on the slopes of the Col du Haag, with Tiesj Benoot and Nicolas Prodhomme setting the pace at the front of the group of favourites.
"Today was a day Paul felt good - he is really good on these medium-hard mountain days," Benoot explained. "He said he felt good, so we tried to launch him on the last climb, and it paid off."
While Yellow Jersey holder Pogacar attacked with 1.5km left on the Col du Haag, Seixas paced himself to drop Ayuso and Florian Lipowitz before catching Vingegaard at the summit.
He went on to finish third on the stage, crossing the line behind Isaac del Toro but ahead of Vingegaard. His reward included four crucial bonus seconds, strengthening his quest for a podium finish.
"This result gives me even more confidence, but the Tour is far from over," Seixas said. "The hardest part is yet to come, and I'm staying pragmatic. There's still so much to do tomorrow."
Sunday's 15th stage promises further challenges as the peloton faces the daunting climb to the Plateau de Solaison, a route Seixas is extremely familiar with.
"It's not far from my grandparents' place, and I've trained there often," he said. "It's one of the toughest climbs I know. If tomorrow goes as well as today, that would be perfect."