FRISCO, Texas -- As he spoke a little more than six hours before Tuesday's trade deadline, Dallas Cowboys owner and general manager Jerry Jones was asked if his team could contend for a Super Bowl.

"I think that we have the potential," Jones said on 105.3 The Fan in Dallas. "I don't think that the way we've played so far will get us there on balance, the good and the bad. Where we are this morning with our health and where we are with how we can play as you go through the position groups, as you go through what we've put together relative to how we're playing defense, our strategies, just our approach. And I look at the competition -- yeah, I think we've got a chance to get in the playoffs and if we do that and we're healthy, I think we've got a chance to advance."

Jones said all of this knowing the Cowboys (4-3) were making a play for New York Jets safety Jamal Adams. The teams had talks the day before the 4 p.m. ET deadline and were in communication up until the final moments.

It would have continued to show an all-in move by Jones, made just a few days after the Cowboys acquired defensive end Michael Bennett from the New England Patriots for a conditional seventh-round pick in 2021.

The Cowboys were willing to part with a first-round pick and a third-day pick or a potential player for Adams, but they were not willing to meet the Jets' ask for first- and second-round picks in 2020 and a second-rounder in 2021, according to sources.

They viewed Adams as a difference-making player in the same way they viewed Amari Cooper last year when they gave up a first-round pick to the Oakland Raiders at the trade deadline for the wide receiver.

Adams would have helped the Cowboys' chances to make Super Bowl LIV, while also giving them a chance to compete in the future since he would have been under contractual control through 2021. And the deal would not have affected the team's ability to retain quarterback Dak Prescott or Cooper with long-term contracts.

"You gotta believe in the coaches. You gotta believe in the organization, the owners, the player personnel guys who make the big decisions," linebacker Jaylon Smith said before the Cowboys' bid for Adams became public. "You gotta believe in what they're doing. So for us, it's just about having a full faith and coming out and doing what we need to do."

The attempt to acquire Adams was a little different. The Cowboys have not put much value on the safety position in terms of free agency or the draft. Their starters are Jeff Heath, who is an undrafted free agent, and Xavier Woods, a 2017 sixth-round pick. They dabbled in Earl Thomas III talks last year with the Seattle Seahawks, but were not willing to give up a first-round pick.

Before the 2019 draft, the Cowboys visited with almost all of the top safeties, and as they were on the clock with the 58th overall pick, the decision came down to defensive tackle Trysten Hill and safety Juan Thornhill. Ultimately, they went with Hill, who has been inactive for three games. Thornhill has started every game for the Kansas City Chiefs and has one interception.

Thornhill is 24. Adams, who is in his third season with the Jets and a defensive captain, is two days older than Thornhill.

A major component of the trade for Cooper was the receiver's age. He was 24 when the Cowboys traded for him and under contract for one more year. Not having a first-round pick in 2019 stung the Cowboys a little and might have stung a little next April, too.

But not too much, if that potential trade for Adams had worked out as well as the Cooper trade.

"The way we looked at it, [Cooper] was such a young player. We loved him coming out, he had been productive in Oakland. He still was 24 years old, so we felt like that was a first-round pick with a guy who there was not a lot of uncertainty about," coach Jason Garrett said. "We felt like he could be a great player and step in and really help our football team, and he has. He had a big impact on our team last year. He's off to a great start this year. So that was one that we were excited to make.

"When you get to the draft and you don't have a first-round pick, you look at each other and say 'Just remember, that guy No. 19 is our first-round pick,' and we did that."

That the Cowboys could not pry Adams away from the Jets does not diminish their chances to make a run with nine games to play.

Bennett, who has 65.5 sacks, gives them another proven pass-rusher who should help the secondary. His production in New England was down because his snap count was low. He was suspended one week because of conduct detrimental to the team, but Patriots coach Bill Belichick gave Bennett solid reviews to Garrett.

DeMarcus Lawrence played with Bennett at the Pro Bowl.

"Just the knowledge with him and the pride over his years, being able to bring that here is a blessing," Lawrence said.

Said Smith: "He played with an edge, tenacity, full-confidence in himself. He's never out there doubting. You need that."

Given the San Francisco 49ers' start to the season at 7-0, and the losses to the New Orleans Saints and Green Bay Packers, the best path to the playoffs for the Cowboys is by winning the NFC East, which is what Jones has acknowledged. With the Seahawks, Los Angeles Rams and Minnesota Vikings having better records at the moment, the wild-card route would be bumpy, although the Cowboys have games remaining against the Rams and Vikings.

"Every moment, every game, every second is precious," defensive end Robert Quinn said. "You don't want to say, 'We've got tomorrow,' because tomorrow is not promised. So why you have a guy like Michael Bennett, a guy [you're] willing to trade for, you can see they didn't wait around. They took advantage of it and brought in another piece.

"At the end of the day, I'm sure I can speak for everyone, it's time to win and not just in the regular season. We know what time it is."