Reporters can’t speak to Sean Payton about the NFL draft for long without the Saints coach issuing them a reminder.

Though countless people obsess over what may happen in the early portions of the event, he’s known to say, his staff and others around the league just as fervently anticipate what occurs in the moments immediately after the seventh and final round.

That’s when players coming out of college whose names weren’t called during the draft become free agents and can sign a contract with any team they wish. Those deals are nowhere near as lucrative as those that go to draft picks, thus opening a market where teams can potentially find valuable contributors at bargain prices.

“That’s an important part of the process,” Payton said while speaking at the league owners meeting in March. “Our scouts and pro personnel people and everyone involved in the process have done a very good job with that, and we’ve been able to secure pretty good football players that way.”

Yet it’s an equally crucial period for NFL agents and their clients, who are barely afforded time to distinguish which offers give players an opportunity to make a roster and which ones are setting them up to be mere camp bodies.

“Negotiations go quick because teams want to get players ready for minicamps and (organized team activities),” said Martin Fischman, a New Orleans-based agent who represents four prospects from Louisiana universities hoping to be chosen in this year’s draft, which runs from Thursday to Saturday. “There’s not a lot of time to make very important decisions.”

The crop of undrafted rookie free agents signed by the Saints last year illustrates those realities well.

One day after the 2013 draft concluded, the Saints announced they’d signed 13 players who’d gone unselected. Seven of those players failed to make New Orleans’ 53-man roster. But, impressively, six of them did: defensive end Glenn Foster, backup quarterback Ryan Griffin, tight end Josh Hill, linebacker Kevin Reddick, offensive lineman Tim Lelito and cornerback Rod Sweeting.

Those six were joined by a seventh undrafted college free agent who wasn’t signed until the conclusion of a rookie minicamp to which tryout players were invited: running back Khiry Robinson.

It’s possible none of those players need any introduction to Saints fans. Other than Griffin, in varying degrees, all manned offensive, defensive or special-teams roles that helped the Saints win 12 of 18 games and reach a divisional playoff for the fourth time since Payton arrived in 2006.

Griffin, the No. 3 quarterback, wasn’t active for any games last season. Still, the Saints found it worth promoting him from the practice squad to the 53-man roster in October when St. Louis tried to sign him away.

Their eventual contributions to the Saints notwithstanding, none of those players got a signing bonus of more than $7,000, which is what Foster got, according to the sports contract website

Hill, Lelito, Reddick and Sweeting reportedly got signing bonuses of $5,000, and Robinson got $1,000, Spotrac said. A media report said Griffin’s initial deal gave him a $5,000 signing bonus.

Compare those figures to what Spotrac said was the signing bonus for wide receiver Kenny Stills, picked relatively late in the fifth round of last year’s draft: $194,452. Even that seems cheap given the fact he later had five touchdown receptions and led the NFL in yards per catch (20).

While the NFL Players Association website indicates all of those players’ rookie-year base salaries were $405,000, and their 2014 one is $495,000, the disparity in signing bonuses hint at why teams covet college free agents with high upside and why players hope to instead be drafted.

“Ninety percent of our coverage is devoted to 10 percent of the draft: the first round,” Payton said at the owners meeting. “(But) that period of time that takes place after the seventh round, which is about two to three hours, is amazingly important.”

In a recent Associated Press story, new Tampa Bay coach Lovie Smith likened that window of time to “what colleges go through with high school players,” with coaches trying to convince prospects facing humble pro beginnings that their program gives them the best chance to succeed in the league.

Adding to that, Fischman — the New Orleans agent representing linebacker Justin Anderson and wideout Darryl Surgent of Louisiana-Lafayette, LSU linebacker Lamin Barrow and Southeastern Louisiana defensive back Todd Washington — teams begin recruiting players they believe may go unpicked ahead of the draft.

He recounted how a team this offseason asked him not for a client’s number — but the client’s mother’s number so they could talk to her about the career benefits their organization could provide her son.

“I had not heard that one yet,” Fischman said of the tactic.

Nonetheless, not every team is as ripe as the Saints are for college free agents to thrive on. While they can appreciate the urgency NFL teams have to get such rookies under contract, both agents and their clients try to avoid organizations where their chances are slim, Fischman said.

“An arm, a leg for minicamp? That’s not our client,” Fischman said. “Our client is not an extra tackling dummy. We want him to have an opportunity to make a team’s roster.”