Dylan Teuns of Belgium and Giulio Ciccone of Italy battled it out for 1st place in the stage.
Jumbo-Visma's George Bennett slots into fourth, two seconds ahead of Thomas who is 49 seconds off yellow.
Overall, Ciccone leads France's Julian Alaphilippe, who wore the yellow jersey in the morning, by six seconds after the Deceuninck-Quick Step rider had to settle for sixth place. Teuns moved up to 3rd.
Defending champion Geraint Thomas was the strongest of the overall contenders on the 160-km route from Mulhouse to La Planche des Belles Filles.
"Then I was told I was first [in GC] so I quickly forgot about finishing the stage second", added Ciccone, who describes himself as a "very quiet guy, on and off the bike".
Now that he has exceeded his own expectations, Ciccone is turning his sights to his day job on the Tour - helping Australian Richie Porte become an overall contender again after crashing out in the last two editions.
Manchester United and Inter €13.5m apart in valuation of Romelu Lukaku
The former champions have a busy pre-season as they travel to Australia, China, Singapore, and then return England. Pogba is featured among the players modelling the kit despite recent links regarding a move away from the club.
Griezmann close to Barca move
Antoine Griezmann will look to complete his Barcelona move in the coming days, with the details of the transfer becoming public. Griezmann has always been expected to join Barcelona since his impending departure from Atletico was announced in May.
Oldest Homo sapiens outside Africa found in Greece
Given the fragmentary nature of the skulls, they were hard to remove and clean, though that eventually happened in the 1990s. A piece of skull found in a cave in Greece suggests our human ancestors left Africa far earlier than anthropologists thought.
Since the Planche des Belles Filles featured for the first time on the Tour in 2012, the rider in yellow after that stage went on to win the race - Bradley Wiggins in 2012, Vincenzo Nibali in 2014 and Chris Froome in 2017.
Thomas and his co-captain Colombian 22-year-old Egan Bernal, who dropped nine seconds to the Welshman, were part of an Ineos team effort that controlled the pace over the most of the course as the peloton crossed seven mountains.
"I was starting to blow though - it was solid - but it was pretty decent". But, you have to take the emotion out of it, the fact that it is the Tour de France, and make a rational decision.
"I just wasn't up to it today, it's a bitter realisation", said Bardet, who has twice finished on the podium but who was the day's biggest loser.
Friday's seventh stage, the longest of the 2019 Tour, returns to flatter terrain as the peloton races from Belfort to Chalon-sur-Saone.