McCarthy finished fifth in Rodez behind stage favourites Matthews, Greg van Avermaet (BMC), Edvald Boassen Hagen (Dimension Data) and Phillipe Gilbert (QuickStep).
McCarthy experienced an eventful race debut with BORA-hansgrohe, which was at the centre of controversy following the disqualification of world champion Peter Sagan in the opening week.
The outfit has clearly regrouped and today repaid faith to the upbeat 24-year-old, who has proved a loyal and effective worker to Sagan and other leaders in the squad here and generally.
“This morning in the meeting the team gave me the opportunity to go for it," McCarthy said. "I’m really happy; okay, [I'm] not on the podium but to show I was strong enough to have a pretty good performance was great."
The punchy 181km race to Rodez proved selective and tough enough to disrupt the general classification with Chris Froome (Sky) back in yellow. The 576m ramp to the finish topped out at almost 10 per cent and saw McCarthy finish one second behind Matthews as part of the elite group.
“I can’t be too disappointed another Australian won. I’m good friends with Michael and I’m happy to see him get the victory. I was hoping to jump up there on the podium but I can be happy with fifth in my first Tour de France,” he said.
“It was a hectic finale and I had to keep my cool. I knew there was one bullet you could use on that final climb and I didn’t want to have to use too much too early," added McCarthy. "I kept it pretty cool the last 40km and put myself in a good position into the final. My teammates helped me with that, so it was really good.”
The Tour is heading toward the Alps but there are more undulating stages on offer, which McCarthy said he may be in a position to vie for.
“I’m always looking for opportunities and Bora wants to get another win before we go,” he said.
“After everything that has happened, I think it would be great to thank the team and sponsors with a podium or victory in the final week of the Tour.”