Teams make their picks in the first round of the NFL Draft based on a calculation factoring need, best player available, and system fit. As the Draft continues, the quality of the prospects goes down. This means talent available takes on more importance. By the sixth and seventh round, teams are looking for some player, any player who might be able to make some kind of a contribution some day. The position isn't important. Most of the prospects are never going to do anything of note in the NFL so finding any kind of contribution gives your team an advantage.
Here are three specific ways teams can try to get a leg up in the sixth or seventh round.
Talented Character Concern: Almost everybody who is available in the sixth round has significant flaws. For most players it is a lack of talent. They don't have the requisite athleticism, or their games are unrefined. Some others might have only performed well at a small school bringing into question whether their production was only a product of their competition. There are a handful of players who would be taken on the first two days if their talent was the only factor but have fallen because of extreme character concerns. Character concerns can easily drop a player out of the first round into the second or third, but there are some major red flags if a team will not touch a player until the sixth.
Somewhere around the fourth or fifth round is the cutoff point to where a team can reasonably expect to find a contributor. If your team does not believe a character concern can be trusted to fly right, it wouldn't make sense to take said character concern when there is a potential contributor available. In the sixth or seventh round, however, these character concerns are up against players whose most likely career trajectory is the practice squad followed by bottom of the roster status followed by no longer being in the NFL. You aren't sacrificing much by rolling the dice on a character risk with these picks.
Last year I was hoping the Jets would throw a sixth or a seventh round pick at Da'Rick Rogers, a former All-SEC wide receiver at Tennessee who left the team after several reported incidents. Rogers went undrafted. He was signed and cut by the Bills before the Colts picked him up. He worked his way into Indianapolis' lineup and had a 6 catch, 2 touchdown game late in the year. How much would he have helped the Jets? I don't know, but I do wonder.
Colt Lyerla, the athletic tight end from Oregon, might be a good example for this year. He pleaded guilty to a drug charge in December and quit on his team last year. Is this a guy a team should touch with a ten foot pole when there are still other potential contributors on the board? Probably not, but in the sixth round you might roll the dice because 6'4" 240 pound men usually do not move like him.
Non-Premium Positions: There are certain positions in the NFL where having premium talent is significantly less important than others. They are positions like kicker, punter, and fullback. Players at these positions help teams win games. Having a bad one can hurt your team. Is there a big difference, however, between having an elite player at these spots and having a functional one? There is not unless we're talking about a generational player like Adam Vinateri.
Because of this, these players are available late. While few players at premium positions pan out in the sixth round, an investment in a non-premium spot can easily leave a team set for five to ten years. If a team also has an expensive veteran at one of these positions, this can free up cap space with a younger, cheaper player.
Ron Wolf Method: The great NFL executive Ron Wolf felt a team should take a quarterback almost every year. The position is the most difficult to evaluate so the more lottery tickets a team buys, the more likely it is to hit. Late in the Draft is a spot to find a potential developmental guy. Something should stand out about him. Maybe his team won a lot of games, but he didn't get credit because he was surrounded by talent. Maybe he was prolific but dismissed as a system quarterback because of playing in an offensive-friendly scheme. Maybe he has a big arm, and his game needs refinement. Young quarterbacks can surprise people with their development so it is not a bad idea to bring in a new one every year.