With their win in Game 6 on Friday, the Flyers won a playoff series for the first time since 2012. It’s an accomplishment that this team can certainly be proud of, given where they were just a season ago.

But that’s not what teams in the hub cities of Toronto and Edmonton really have their sights set on in these playoffs. It is about a greater goal. So while the Flyers won the Round Robin and claimed the top seed and then won their first-round series, they don’t plan on making a second-round exit.

With that in mind, the Flyers were given a tough test in an upstart Montreal Canadiens team that had nothing to lose in the playoffs and played really hard, tight-checking hockey. The Flyers next opponent, the New York Islanders, plays a very similar style. So as the team reflected on their first-round series win and looked ahead to round two, there was a lot more self evaluation than looking at the opposition.

The Flyers didn’t feel they played close to their best in the first round, and they know they can’t get by on that in the next one.

“I do,” GM Chuck Fletcher said when asked if the team has another level to reach with their play. “We’ll probably have to find it if we want to continue to win games and move on. You have to give a lot of credit to the Montreal Canadiens. They played a very structured, very fast game. They put a lot of pressure on us. I think we put a lot of pressure on them at times too. There wasn’t a lot of free ice out there. We had to battle for everything we had. I do think we started to get a few more chances offensively as the series went on. I think our power play got a little better as the series went on. There’s no question, I think there’s another level we need to get to and that probably goes for every team that’s still playing.”

Alain Vigneault agreed with Fletcher’s assessment, noting that some of the team’s top players still have more to give.

“Yes, there is and Chuck’s 100 percent right. I do believe we have another level, another gear that we can get to,” Vigneault said. “Most of it has to do with our execution, our plays, and our ability to make plays with the puck.

“We have some real skill players, some real good players on our team. I understand going through the process, the pressure that comes with wanting to win. We were able to win that first round, playing hard hockey, good hockey. Still when you analyze it, I really believe that there’s another level that we can attain if our top end guys execute the way we have seen them execute throughout the year. I don’t want to get into specific names, but there are a few guys that need to get that execution, those plays that you need to make under pressure, those plays that you need to make to get the puck out of our end quicker, those plays that you need to make through that neutral zone, and those plays in the offensive zone that a skilled player can make. Our guys did, but didn’t do it with the regularity and consistency that you need, especially you need at this time of the year.”

Two of those top players, Sean Couturier and Travis Konecny, are probably among the group that Vigneault wants to see more from in the next series. The group of Couturier, Konecny, Kevin Hayes, Claude Giroux and James van Riemsdyk combined for 109 goals in the regular season. In the series against the Canadiens, they combined for one goal, with Hayes scoring his first in the deciding Game 6.

“I think we all know there’s another level that we can get to,” Konecny said. “Individually and as a team, everybody wants to step up and do a little bit more. I think it’s just going to come with time. That series brought us together a little bit more and more as we were going through it. Areas we can work on, just cleaning up little mistakes here and there in the D-zone, getting a little cleaner will definitely help us transition into our speed game. I think that’s going to be key.”

“I think it’s important just to get better and better as the playoffs go,” Couturier said. “Could have been easy to just get frustrated and try to go on our own page. We stuck together through adversity in that first round and came out on top. Now we just have to stick together again and build off that first round. Obviously find that other level.”

For Konecny in particular, he led the team with 24 goals and 61 points in the regular season, but had just one assist in the six-game series. While the Flyers will be looking for more offense from him, Konecny is focused on doing what he can to help the team win games.

“A win’s a win in the playoffs the way I look at it,” Konecny said. “As long as we’re doing well, if I’m not on the scoreboard, it doesn’t really matter. I’m trying to contribute in other ways. There’s lots of ways to win a hockey game. It doesn’t just involve points and scoring. I’m aware I’ve got to step up a little bit more. We’ll just move forward now and focus on the next series.”

The next series, the second round against the Islanders starting on Monday night, is the deepest the Flyers have been in the playoffs since 2012. Four members of the current roster were part of that team and had to endure the long eight-year drought without a series win. For two of those members, Giroux and van Riemsdyk, that was their third straight season with a taste of extensive playoff hockey.

Since then, that group of players has a lot of hockey under their belts, and it makes Fletcher’s words about taking advantage of the opportunity that’s in front of the Flyers more powerful than ever.

“You only get so many chances in this business. There’s 23 teams that are not playing and eight of us that still are. You always like to think you are going to get this far every year, but it’s tough,” Fletcher said. “We got to take advantage of this opportunity. It’s a great learning experience on the one hand for a lot of our young players that haven’t played a lot of playoff hockey or haven’t played any playoff hockey until this year. Opportunities are not always there and we have a good one right now.”

Kevin Durso is Flyers insider for 97.3 ESPN and Flyers editor for SportsTalkPhilly.com. Follow him on Twitter @Kevin_Durso.